Methodologies linking patterns from multi-modality datasets

ABSTRACT

A method is disclosed to acquire imaging and non-imaging datasets from like objects. A linkage is found using a partial least squares (PLS) technique between imaging and non-imaging datasets. The linkage is then reduced to an expression of a single numerical assessment. The single numerical assessment is then used as an objective, quantified assessment of the differences and similarities between the objects. The data each dataset can be aspects of performance, physical characteristics, or measurements of appearance.

CROSS REFERENCES TO RELATED APPLICATIONS

This application is a continuation of and claims the priority benefit ofU.S. patent application Ser. No. 11/242,820 filed on Oct. 3, 2005, whichclaims priority to U.S. Provisional Application No. 60/615,767 filed onOct. 4, 2004, titled “Neuroimaging Methods and Systems”, the entirecontents of which are incorporated herein by reference

FIELD OF INVENTION

This invention relates to imaging, and more particularly relates togeneral computational mathematical methodologies linking multi-modalityimaging and non-imaging datasets for valuating an effect upon objectsfrom which data in the datasets is obtained, and most particularlyrelated to biomathematical methodologies linking multi-modalityneuroimaging and non-imaging datasets for characterizing patient groupdifferences and for valuating the efficacy of treatments forneurological, psychiatric, and related disorders upon human subjectsfrom whom data in the datasets is obtained.

BACKGROUND

Neuroimaging researchers frequently acquire multi-modality image dataand various non-imaging measurements. For example, FDG-PET andstructural (e.g., volumetric) MRI brain images as well as a completebattery of neuropsychological tests are acquired from each healthysubject every two years in our NIH sponsored longitudinal APOE-ε4 study.In their study of imaging neurofibrillary tangles and beta amyloidplaques using 2-(1-[6-[(2-[18F]fluoroethyl)(methyl)amino]-2-naphthyl]ethyli-dene)Malononitrile(FDDNP) (Shoghi-Jadid, K. et al. 2002), Researchers from UCLA acquiredtriple imaging datasets, FDG-PET, FDDNP-PET and T1 weighted volumetricMRI. Similarly, Researchers at the University of Pittsburgh used dualPET tracers, FDG and PIB in their study of imaging brain amyloid in AD(Klunk, W. E. et al. 2004). The availability of multi-modality imagingdatasets provides researchers an opportunity to examine multi-processessimultaneously and yet poses a methodological challenge in having themulti-datasets optimally integrated and utilized for the understandingof the underlining biological system.

There have existed methods that make use of data from one image modalityfor the analysis of another. People have long used image fusiontechnique for localizing functional findings with the anatomical mapprovided by structural images (as an example, see (Reiman, E. M. et al.2004)). Similarly, region of interest (ROI) defined on the anatomicalimages can be used to extract data from functional dataset toinvestigate experimental condition manipulated brain responses. Takingthe advantage of high resolution, volumetric MRI has also been routinelyused to correct the combined effects of partial volume average andatrophy related to the functional images (Pietrini, P. et al. 1998). Inthe FDG-PET study, this correction allows researchers to determine ifthe underlining cause of the observed brain functional alternations ispurely glucose metabolic pathway or mostly the structural relate(Reiman, Chen, Alexander, Caselli, Bandy, Osborne, Saunders, and Hardy2004). Besides these procedures listed here and used in mostlystructural-functional studies, findings from one imaging modality areoften correlated with the that from another imaging modality or fromnon-imaging measurement using conventional correlation analysis(Shoghi-Jadid, Small, Agdeppa, Kepe, Ercoli, Siddarth, Read,Satyamurthy, Petric, Huang, and Barrio 2002). Overall, the approacheslisted here are relative straightforward and mostly in the context ofanalyzing primarily the data from one single-modality using another,supportive and secondary. In contrast, our approach proposed in thecurrent study, multi-modality, inter-networks and multivariate innature, is to establish the optimal way to link multi-datasets and tocombine the information from each of the datasets for enhancingresearcher's ability to detect alternations related to the experimentalconditions or the onset, progress or treatments related to the study ofdiseases.

As mentioned above, our approach will be multivariate in nature.Multivariate analysis has been long used in single-modality studiescomplementary to univariate analysis. These single-modality,intra-network and multivariate analysis, model-based or data-driven, areto characterize brain inter-regional covariances/correlations. Thesemethods, voxel- or ROI-based, included principal component analysis(PCA) (Friston, K. J 1994), the PCA-based Scaled Subprofile Model (SSM)(Alexander, G E and Moeller, J R 1994), independent component analysis(McKeown, M. J. et al. 1998; Duann, J. R. et al. 2002) (McKeown, Makeig,Brown, Jung, Kindermann, Bell, and Sejnowski 1998; Arfanakis, K. et al.2000; Moritz, C. H. et al. 2000; Calhoun, V. D. et al. 2001; Chen, H. etal. 2002; Esposito, F. et al. 2003; Calhoun, V. D. et al. 2003;Schmithorst, V. J. and Holland, S. K. 2004; Beckmann, C. F. and Smith,S. M. 2004) and the Partial Least Squares (PLS) method (McIntosh et al.1996; Worsley, K. J. et al. 1997). Also included are Multiplecorrelation analysis (Horwitz, B 1991; Horwitz, B. et al. 1999),structure equation model (Mcintosh, A. R. and Gonzalez-Lima, F 1994;Horwitz, Tagamets, and McIntosh 1999), path analysis (Horwitz, B. et al.1995; Worsley, K. J. et al. 1997), and dynamic causal modeling (Friston,K. J. et al. 2003). These methods have typically been used tocharacterize regional networks of brain function (and more recentlybrain gray matter concentration (Alexander, G et al. 2001)) and to testtheir relation to measures of behavior. No one of these multivariatemethods, however, has been used to identify patterns of regionalcovariance among multi-imaging datasets.

Motivated by the availability of the multi-neuroimaging datasets andencouraged by the success of single-modality network analysis,especially the PLS works, we set out searching for tools that allow usto seek for the maximal linkage among the multi-datasets or to optimallycombine them for increased statistical powers. We believe dual-block PLS(DBPLS) as well as multi-block PLS (MBPLS) should be the first set oftools we would like to explore for such purpose. We will list thechallenges and difficulties in performing inter-modality analysis usingPLS and our very own plan for further methodological development later.First, however, a review is in demand for the general PLS methodology,the success of DBPLS in the neuroimaging field (mainly by McIntosh andhis colleagues) and that of MBPLS mainly in the chemometrics andbioinformatics areas.

Review of the PLS Method

Citing from the Encyclopedia for research methods for the socialsciences, PLS regression is a relative recent technique that generalizesand combines features from PCA and multiple regressions. It isparticularly useful when one needs to predict a set of dependentvariables from large set(s) of independent variables (Abdi, H. 2003).

The traditional use of PLS regression is to predict (not to link)dependent dataset Y from c (c≧1) independent datasets X₁, . . . X_(c),hence the term of PLS regression. Note that in this writing thevariables in each dataset are arranged column-wise in the data matrix.In addition to the PLS regression, we are also interested in its use todescribe the linkages among multi-dataset without the labeling ofdependent or independent. With details of the PLS linkage methodologydevelopments to be described later, we provide here a review of the PLSregression methodology. In a sense, PLS is not needed when Y is a vector(single variable dataset) and X is full rank (assuming c=1) as the Y-Xrelationship could be accomplished using ordinary multiple regression.For our neuroimaging studies, especially our inter-network analysis, thenumber of voxels/variables is greater than one, and in fact much largerthan the number of subjects/scans, multicollinearity exists for eachdataset. Several approaches have been developed to cope with thisproblem when Y is a vector (which is not the case in our neuroimagingstudy). The approach, called principal component regression, has beenproposed to perform a principal component analysis (PCA) of the X matrixand then use the principal components of X as regressors on Y. Thoughthe orthogonality of the principal components eliminates themulticollinearity problem, nothing guarantees that the principalcomponents, which explain X, are relevant for Y (Abdi 2003). Bycontrast, PLS regression finds components from X that are also relevantfor Y. Specially, PLS regression searches for a set of components thatperforms a simultaneous decomposition of X and Y with the constraintthat these components explain as much as possible of the covariancebetween X and Y (Abdi 2003). The procedure of finding the first PLSregressor is equivalent to maximize the covariance between a linearcombination of the variables in Y and a linear combination of thevariables in X (the paired linear combinations are referred to as thefirst latent variable pair). This maximal covariance is symmetrical forY and X for this first latent variable pair. Symmetry here is referredto as the irrelevancy of the fact which dataset is designated asdependent. The symmetry is lost for subsequent latent pairs, however, asis demonstrated below.

DBPLS Algorithm:

As mentioned above, DBPLS uncovers the sequential maximal covariancebetween two datasets by constructing a series of latent variable pairs.Starting from original data matrices X and Y (with standardizationnecessary), the first latent variable pair is constructed as follows.The latent variable of X is t=Σw_(i)x_(i) where w_(i) is scalar, and x,is the i^(th) column of X (i=1, 2, . . . ). In matrix form, t=Xw wherew=(w₁, w₂, . . . )^(T) with ∥w∥=1. Similarly, the Y latent variable canbe expressed as u=Yc (∥c∥=1). In the context of dual-imaging datasetsand for matter of convenience, we will refer w and c as singular imageof X and Y respectively. The covariance of the two latent variables, tand u, is therefore cov(t,u)=w′X′Yc (assuming zero mean for variables inboth datasets). The maximal covariance value with respect to w and c canbe proven to be the square root of the largest eigenvalue of the matrixΩ=[X′YY′X] with w being the corresponding eigenvector of Ω, and c beingthe corresponding eigenvector of Y′XX′Y. Prior to the second latentvariable pair, the effects of the first latent variable pair needs to beregressed out from X and Y, referred as deflation in the chemometricsPLS literature:

Express and

${p_{1} = \frac{X^{\prime}t}{{t}^{2}}},{q_{1} = \frac{Y^{\prime}u}{{u}^{2}}},{r_{1} = \frac{Y^{\prime}t}{{t}^{2}}}$

and calculate new X₁ and Y₁ as X₁=X−tp₁′ Y₁=Y−tr₁′

The same calculating procedure will then be repeated for the new X₁ andY₁ matrix pair to construct the second latent variable pair. The thirdand remaining latent variable pairs (up to the rank of X) will becalculated similarly.

MBPLS Algorithm:

The calculation of MBPLS is based on the DBPLS procedure describedabove, with some kind scheme of deflation to take care of the presenceof more than one independent block. Westerhuis et al described thefollowing numerical procedure (Westerhuis, J. A. and Smilde, A. K.2001):

-   -   1, Calculate the first latent variable pair of the DBPLS model        between X=[X₁, . . . , X_(c)] and Y. The scores t and u, weight        w and loadings p and q are obtained. From these, the multiblock        PLS block weights w_(b), the super weights ws and the block        scores t_(b) are obtained.    -   2, w_(b)=w(b)/∥w(b)∥²    -   3, t_(b)=X_(b)w_(b)    -   4, ws(b)=t_(b) ^(T)u/u^(T)u    -   5, ws=ws/∥ws∥²    -   Block score deflation    -   6a, p_(b)=X_(b) ^(T)t_(b)/t_(b) ^(T)t_(b)    -   7a, E_(b)=X_(b)-t_(b)p_(b)    -   8a, F=Y−tq    -   Super score deflation    -   6b, E_(b)=X_(b)−tp(b)^(T)    -   7b, F=Y−tq        For additional components, set X=[E₁, . . . , E_(c)] and Y=F and        go back to step 1.

Different deflation step can be used playing a crucial part in MBPLScalculation. The block score deflation, suggested by Gerlach andKowalski (Gerlach, R. W. and Kowalski, B. R. 1979), led to inferiorprediction. Westerhuis et al. showed that super score deflation gave thesame results as when all variables were kept in a large X-block and aDBPLS model was built. The super scores summarize the informationcontained in all blocks, whereas the block scores summarize theinformation of a specific block. However, the super score deflationmethod mixes variation between the separated blocks and therefore leadsto interpretation problems. In order to overcome the mixing up of theblocks, deflating only Y using the super scores was proposed (Westerhuisand Smilde 2001). This leads to the same predictions as with super scoredeflation of X, but because X is not deflated, the information in theblocks is not mixed up.

Review of DBPLS in the intra-modality neuroimaging studies

McIntosh and his colleagues first introduced DBPLS into the neuroimagingfield in 1996 (McIntosh, Bookstein, Haxby, and Grady 1996) for theintra-modality spatial pattern analysis in relationship to behavior orexperimental conditions. Consequent to this study, Worsley considered analternative PLS procedure, what he referred to as the orthonormalizedPLS (Worsley, Poline, Friston, and Evans 1997) to account for the issueof being invariant to arbitrary linear transformations. Ever since,DBPLS works have been extended, improved and introduced extensively tovarious brain studies mainly by McIntosh and his group. Their effortsincluded further methodological developments such as the extension fromPET to functional MRI studies, from the original PLS to seed-PLS(McIntosh, A. R. et al. 1999) or spatiotemporal-PLS (Lobaugh, N. J. etal. 2001; Lin, F. H. et al. 2003) and numerous applications in brainfunction/disease studies (McIntosh, A. R. 1998; McIntosh, A. R. 1999;Rajah, M. N. et al. 1999; O'Donnell, B. F. et al. 1999; Anderson, N. D.et al. 2000; Iidaka, T. et al. 2000; Lobaugh, West, and McIntosh 2001;Nestor, P. G. et al. 2002; Keightley, M. L. et al. 2003; Habib, R. etal. 2003). Another significant contribution from McIntosh's group is theintroduction of the non-parametric inference procedures, permutation orBootstrapping for intra-modality PLS neuroimaging studies (for example,see the initial introduction paper (McIntosh, Bookstein, Haxby, andGrady 1996)).

Review of DBPLS in the Inter-Modality Neuroimaging Studies

Presented on the World Congress on Medical Physics and BiomedicalEngineering at Sydney, Australia in 2003 (Chen, K et al. 2003), ourgroup reported the inter-network preliminary results linking FDG-PET toMRI segmented gray matter overcoming a huge computing obstacle relatedto the size of the covariance matrix between two imaging datasets(number of voxel in one image data set x the number of voxels inanother). Our aim is to seek direct linkage or regression betweendual-modality imaging datasets (MBPLS regression or MBPLS linkageanalysis).

One year later, researchers from McIntosh's group reported alternativeapproaches for analyzing multi-modality imaging data at 13th AnnualRotman Research Institute Conference Mar. 17-18, 2004 (Chau, W et al.2004). They used the same operational procedure as in theirintra-modality PLS studies in attempting to answer the same question:the experimental condition or behavior related neuroimaging covaryingpatterns. In other words, the roles of neuroimaging datasets are onlyand always the X's blocks in the PLS regression notation above with theexperimental conditions or behavior data as the dependent Y block (Chau,Habib, and McIntosh 2004). Since the direct linkage between/amongmulti-modality datasets is not the purpose of their investigation, thereexist no needs to computationally deal with the issue of the covariancematrix sizes. Also, since the number of X blocks is more than one,investigation on the deflation scheme is needed, but not was consideredin their study.

Review of DBPLS and MBPLS in Chemometrics and Bioinformatics

Though the successes of the DBPLS in the neuroimaging field have beenindeed impressive, the application of MBPLS in this field is yet to bematured, its success demonstrated, and new algorithms developed.Numerous successful applications of both DBPLS and MBPLS, however, havebeen reported in the field of fermentation and granulation for food orpharmacological industries. The importance of PLS in Chemometrics fieldis evidenced by the online editorial in the Journal of Chemometrics(Hiskuldsson, A 2004). An incomplete MBPLS review in these fields isprovided here together with some discussion on their relevance to ourintended neuroimaging applications.

Esbensen at al., analyzed data of the electronic tongue (an array of 30non-specific potentiometric chemical sensors) using PLS regression forqualitative and quantitative monitoring of a batch fermentation processof starting culture for light cheese production (Esbensen, K. et al.2004). They demonstrated that the PLS generated control charts allowdiscrimination of samples from fermentation batches run under “abnormal”operating conditions from “normal” ones at as early as 30-50% of fullyevolved fermentations (Esbensen, Kirsanov, Legin, Rudnitskaya,Mortensen, Pedersen, Vognsen, Makarychev-Mikhailov, and Vlasov 2004).Relevant to our proposal, this study is a clear demonstration of theMBPLS prediction power based on multi historical datasets, the powerthat a physician dreams to duplicate for early diagnosis of a disease.

In another study (Lopes, J. A. et al. 2002), the performance of anindustrial pharmaceutical process (production of an activepharmaceutical ingredient by fermentation) was modeled by MBPLS. Withthe multiblock approach, the authors were able to calculate weights andscores for each independent block (defined as manipulated or qualityvariables for different process stage). They found that the inoculumquality variables had high influence on the final active productingredient (API) production for nominal fermentations. For thenon-nominal fermentations, the manipulated variables operated on thefermentation stage explained the amount of API obtained. As demonstratedin this study, the contributions of individual data blocks to the finaloutput can be determined. The neuroimaging analog of their study is touse PLS to evaluate the relative contribution of various datasets (MRI,FDG-PET, neuro-psychological tests) in accurately predicting the onsetof AD or in evaluating the effects of treatments.

Hwang and colleagues discussed the MBPLS application to the field oftissue engineering in one of their recent publications (Hwang, D. et al.2004). They used MBPLS model to relate environmental factors and fluxesto levels of intracellular lipids and urea synthesis. The MBPLS modelenabled them to identify (1) the most influential environmental factorsand (2) how the metabolic pathways are altered by these factors.Moreover, the authors inverted the MBPLS model to determine theconcentrations and types of environmental factors required to obtain themost economical solution for achieving optimal levels of cellularfunction for practical situations. The multi datasets (or multi-groupsas referred by them) included the group of environmental factors and Cgroups, each of them consisting of a number of metabolites and fluxesthat have similar metabolic behaviors. Like the one by Lopes et al.,this study illustrates the power of MBPLS to assess the relativeimportance of each independent dataset in predicting the behavior ofinterests. Moreover, this study showcases the use of MBPLS to determinethe variable combinations that give rise to the optimal level of thedependent variables.

Note that the MBPLS applications reviewed above are all in the frameworkof multiple-independent (predictor) blocks and a single dependent block,all consisting of no more than N number of variables, where N is aten-thousand times smaller than the number of voxels/variables in theneuroimaging datasets.

Relative to neuroimaging, a major challenge to the multivariate analysisof regional covariance with multiple imaging modalities is the extremelyhigh dimensionality of the data matrix created by including relativelyhigh-resolution neuroimaging datasets. What is needed is a strategy tomake computation of high dimensional datasets using multivariate methodsfeasible.

SUMMARY

Mathematical methodologies are disclosed to find a linkage betweenimaging and non-imaging datasets. The linkage is used to findrelationships among datasets, to combine, summarize information frommulti-datasets, and to construct new numerical surrogate markers forincreased statistical power in the evaluation of the status of objects,both manmade and biological, such as for the evaluation of humans andpossible early treatment and prevention strategies in the fights againsta disease (such as Alzheimer Disease).

Implementations disclose a request to acquire a plurality of datasetsfrom each of a plurality of objects. A linkage exists between thesedatasets, where each dataset is potentially a different modality (e.g.,imaging and non-imaging datasets). The linkage between datasets can befound using a partial least squares (PLS) technique, including DualBlock (DB) PLS or Multi-block (MB) PLS with the conventional criterionor the one established as disclosed herein. Moreover, we also disclosureother analytical techniques for finding the linkage. The linkage is thenreduced to an expression of a single numerical assessment.Alternatively, the linkage can be reduced to a unique solution that canbe characterized by several numbers for each of the assessed modalities.

The single numerical assessment is then used as an objective, quantifiedassessment of the differences and similarities between the objects. Thedata in the plurality of datasets, as mentioned, can be acquired eitherby an imaging modality or a non-imaging modality. The data in eachdataset can be an index, such as an aspect of performance, a physicalcharacteristic, a measurement of appearance, or numerical representationof the inner status of the objects such as the glucose uptake rates/graymatter concentrations from various human brain regions.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete understanding of the implementations may be had byreference to the following detailed description when taken inconjunction with the accompanying drawings wherein:

FIG. 1 shows close correlation between each of 10 possible pairs in a 5block PLS analysis using our newly defined criteria which do not needlabels of dependent or independent datablocks;

FIG. 2A illustrates the use of Partial List Squares (PLS) todiscriminate 14 from 15 young subjects in a preliminary PLS validationstudy, where the x-axis is the PET-PLS subject scores, and the y-axis isthe MRI-PLS subject scores;

FIG. 2B shows the first singular images for PET on the left panel andfor MRI on the right panel for the same preliminary PLS validationstudy;

FIG. 3 illustrates the potential use of a PLS method, where one isinterested in investigating a debatable relationship between mothers'fitness and daughters' cognitive skills, where the X matrix X lists themothers' physical characteristics measurements, and the Y matrix liststhe daughter's cognitive skill measurements;

FIG. 4 demonstrate a procedure to form matrix X and Y for J humansubjects' dual FDG-PET and MRI data, where each row of X/Y is for onesubject's MRI/PET data rearranged as a row vector;

FIG. 5 shows the singular images (MRI on the right panel and PET on theleft panel) generated by first applying data dimension reductiontechnique (PCA) followed by PLS procedure, where it can be noted thatthere is a striking similarity between the PLS with a power algorithmand the present one;

FIG. 6 depicts the use of the PLS derived subject scores as adiscriminator in a jackknife procedure which excludes one subject at atime, and then predicts the membership of the subject using theinformation from the remaining subjects;

FIG. 7 shows the close result SVD relationship calculated by the exactSVD computation procedure which is with high computational costs and bythe proposed method which uses dimension reduction technique first(inexact), where the inexact SVD results is shown on the Y axis, and theexact SVD is shown on the X axis;

FIG. 8A shows respective MRI and PET univariate SPM findings (T-map) inthe preliminary human brain studies, where the results were obtained viaSPM voxel-wise univariate analysis (PET and MRI separately), and thoughwith increased power, the PLS results are compatible with these SPMfindings;

FIG. 8B shows SPM dual-variate findings (F-map) in the preliminary humanbrain studies, where the results were obtained via SPM voxel-wisemultivariate (in this case, each voxel has two variables), and thoughwith increased power, the PLS results are compatible with these SPMfindings;

FIG. 9A shows the voxel-by-voxel PET correlation with the overall MRIPLS latent variable for the preliminary human brain study, where thelatent variable is representative of the overall MRI data, and thus, thePET pattern displayed in this FIG. 9A reflects that PET voxel-by-voxelvariations are related globally to the brain structure measures by MRI;

FIG. 9B shows the voxel-by-voxel MRI correlation with the overall PETPLS latent variable for the preliminary human brain study, where thelatent variable is representative of the overall PET data, and thus, theMRI pattern displayed in this FIG. 9B reflects that MRI voxel-by-voxelvariations are related globally to the brain functions measures by PET;

FIG. 10 shows the PLS singular images obtained by the non-agnosticpartial least squares processing, where this is for the preliminary PETand MRI human brain studies, where similarities of the singular imagesare apparent in comparison to the results of the agnostic approach asseen in FIG. 2B;

FIGS. 11-14 show respective exemplary processes for potentialapplication of a proposed method.

DETAILED DESCRIPTION

Mathematical methodologies are developed and implemented to seek linkagefirst between dual-modality and then extended to among multi-modalityneuroimaging and non-imaging datasets. The methodologies allowresearchers to find relationships among datasets, to combine, summarizeinformation from multi-datasets, and to construct new numericalsurrogate marker of neuroimaging for increased statistical power in theevaluation of possible early treatment and prevention strategies in thefights against a disease (such as Alzheimer Disease).

The idea of multi-modality inter-network analysis using partial leastsquare (PLS) technique. Our group is the first suggesting to investigatethe direct linkage among multi-imaging datasets, and to combineinformation from multi-datasets for increase statistical power with theuse of PLS.

The numerical strategy to make the calculation of PLS with covariancematrix of huge size feasible on a personal desktop/laptop computer. Wecome out the way to divide into small pieces a huge matrix that nocomputer can just simply hold it in memory. Thus, the computation theinter-network PLS becomes feasible on a modern desktop/laptop computer.See C.1 Implementation of DBPLS for voxel-based neuroimaging data.

The special application of the algorithm to seek covarying patternsamong multi-modality neuroimaging datasets for the study of Alzheimerdisease (AD), risk of AD, evaluation of early treatment or prevention ofAD. We propose to look the covaried-pattern changes acrossmulti-modalities, and to use latent variable pairs as multivariate index(indices) for the calculation of statistical power via Monte-Carlosimulation and believe the indices are with improved sensitivity andwithout the need to correct for multiple comparisons. See C.2.Assessment of Statistical Significance, D. 1 and D. 3

The idea and application of the algorithm to seek covarying patternsbetween imaging and non-imaging datasets as a tool for diagnosis. Weproposed to generalize our initial PLS AD application to other diseasediagnosis such breast cancer with mammography and breast MRI, and toother research areas such as the search of linkage between neuroimagingdata and genomic information. See III.3 of A, specific aims. D. 1.5

The re-definition of the multi-block PLS as a procedure to seekcovarying patterns among all blocks without designating one block asdependent block and others as independent blocks. The conventional PLSis in the frame work of predicting a single dependent block based on theobservation of one or more independent blocks. We ask the question ofseeking relationship among inter-dependent multi-blocks. With thatredefinition, we introduce various alternative object functions andalgorithms to seek the solution of the newly defined multi-block PLS.See D.4.1

The theoretical findings of mathematical and theoretical discussions onthe multi-block PLS re-definition for the existence and uniqueness ofits solution. We will discuss conditions under which there exists oneand only one solution for the newly introduced object function. See theAppendix.

The applications of the newly defined multi-block PLS approach to studythe inter-network relationship between multi-modality neuroimagingdatasets especially in the study of AD.

The conceptual introduction of inter-system independent componentanalysis (inter-ICA) and inter-system structural equation modeling(inter-SEM). ICA has been widely used in constructing a set ofstatistical independent components for a single dataset (one system). Weput forward the idea of inter-system ICA and proposed algorithm to havethat realized. Similar comments can be made for inter-system SEM. See D.4.3

A. Specific Aims

The overall goal is to develop multivariate analysis algorithms foranalyzing multi-modality neuroimaging and non-imaging datasets in asystematic inter-network approach. With this analytical tool, we aim toa) study the linkage among imaging/non-imaging datasets, b) toinvestigate relevant importance of each dataset, as a whole, incontributing to the predictability of brain functions, brain diseaseonset, clinical outcomes in general or treatment evaluation, and c) toapply the developed algorithms to various neuroimaging studies includingespecially our longitudinal study of the genetic risks of Alzheimerdisease (AD) associated with the apolipoprotein ε4 (APOE-ε4) alleles.

Two versions of partial least square (PLS) technique, dual-block PLS(DBPLS) and multi-block PLS (MBPLS) will be investigated for ourinter-modality methodology endeavor. It is worth to note that DBPLS hassuccessful applications in intra/single-modality neuroimaging studies(McIntosh, A. R. et al. 1996).

The specific aims can be categorized as methodological developments,general biomedical applications and the special application to ADneuroimaging studies:

I: Answer the Methodological Development Challenges

I.1, Developing/Implementing inter-modality and voxel based or region ofinterest based DBPLS/MBPLS algorithm. Strategies will be planned toovercome the difficulty associated with the extremely high size of thedual-imaging dataset covariance matrix.

I.2, Assessing the statistical power and type-I errors of the PLSuncovered inter-modality networks. Efficient non-parametric procedureand Monte-Carlo simulation will be proposed for such purposes.

I.3, Maximizing the linkage strength among datasets based on newlyproposed MBPLS object functions. In addition to the fact that MBPLS onlyseeks the maximal linkage between the dependent dataset and the set ofindependent datasets, we propose to investigate the simultaneousmaximization of covariances (or other index of linkage strength) of eachof all possible dataset pairs.

II: Answer the Data Analysis Challenges in General BiomedicalNeuroimaging/Non-Imaging Studies

II.1, Providing a tool to study the inter-network relationship amongmulti-datasets. With this tool, for example, one could examine how theglucose uptake pattern over various brain regions measured by F-18fluoro-2-deoxyglucose (FDG) and positron emission tomography (PET) isrelated to the spatial distribution of segmented gray matter volumemeasured by magnetic resonance imaging (MRI). Another example, one canuse this tool to study the global pattern linkage among cerebral glucoseuptake (by FDG-PET), the distribution of amyloid plaques (byN-methyl-[C-11]2-(4′-methylaminophenyl)-6-hydroxybenzothiazole(PIB)-PET), and gray matter spatial pattern (by MRI).

II.2, Making available a means to combine and integrate the informationfrom multi systems/datasets for increased statistical power fortreatment evaluation, risk assessment or clinical diagnosis.

II.3, Offering a procedure to assess relative importance of each datasetin predicting the clinical outcomes and in evaluating treatments.

II.4, Initiating our efforts to study other mathematical algorithms forinter-network relationship in addition to PLS. Among the alternativemulti-dataset analysis tools are the inter-network independent componentanalysis (ICA) and the inter-network structural equation modeling (SEM).

III: Answer the Data Analysis Challenges Especially in the NeuroimagingStudies of Ad and AD Risk, and in Other Medical Research Areas

III.1, developing a prediction scheme using cross sectional andlongitudinal FDG-PET and MRI data (and possibly together withneuropsychological data) to assess the risk for the symptomatic onset ofthe clinical AD for healthy individuals carrying 0, 1 or 2 copies ofAPOE-ε4 allele (PLS application to data acquired with the support ofNIMH MH057899-06)

III.2, constructing a clinical diagnostic scheme using cross sectionaland longitudinal FDG-PET and MRI data (and possibly together withneuropsychological data) to calculate the probability and average timeduration an MCI patient converts to AD

III.3, exploring the possibility of applying PLS to other medicalresearch and clinical areas where the multi-modality [non-]imagingdatasets need to be combined or evaluated. An example would be themammography and breast MRI.

Methodological Challenges

With our specific aims in mind, our focuses will be to propose,implement and evaluate strategies to conquer challenges listed here.

I. Demonstrate the need and the power of the inter-network analysis inneuroimaging studies. Please see the Significance session and thePreliminary Results session for details.

II. Make the computation feasible for the inter-network neuroimaging PLSanalysis

To illustrate the computational challenge using dual-imaging datasetDBPLS as an example, let us consider the size of images used in atypical Statistical Parametric Mapping (SPM) PET analysis with 2 mmcubic voxel. Note that Statistical Parametric Mapping refers to theconstruction and assessment of spatially extended statistical processesused to test hypotheses about functional imaging data, such by the useof software also called SPM (http://www.fil.ion.ucl.ac.uk/spm/). SPMsoftware can be used for the analysis of brain imaging data sequences.The sequences can be a series of images from different cohorts, ortime-series from the same subject. SPM software can be for the analysisof fMRI, PET, SPECT, EEG, and MEG.

The number of brain voxels could be 246,082 (almost a quarter millions).If this number is for both datasets in a dual-network PLS analysis, thenthe covariance matrix will be 246,082 by 246,082. The same calculationdifficulty exists for the newly introduced object function in this study(more on this below). Implementation feasibility is important not onlyfor the inter-network PLS itself, but more for its type-I error andstatistical power computation (below).

III. Further develop PLS procedure in answering challenges in theneuroimaging research area.

Type-I error and statistical power assessment: For assessing thestatistical significance (type-I error) of various aspects related tothe uncovered covarying patterns and correlations between two datasetsor among multi-datasets, non-parametric approaches such as Jacknife,bootstrap and permutations (permissible with experimental designs)resampling techniques as well as Monte-Carlo simulations will be adoptedwhich, except our own proposed Monte-Carlo simulation, have beendemonstrated their usefulness in the DBPLS intra-modality neuroimagingstudies (McIntosh, Bookstein, Haxby, and Grady 1996). Moreover, in orderto propose a single PLS index or a set of PLS indices as surrogatemarker in designing new studies such as treatment evaluations, we needto calculate the statistical power.

New MBPLS linkage strength indices establishment: For the MBPLS casewhere no meaningful dependent/independent labels can be assigned to eachdataset, we intend to institute a new object function in finding themaximum linkage strength among these multi-datasets. This is in additionto the MBPLS algorithm that seeks the maximal linkage between thedependent dataset and the set of independent datasets. We willinvestigate the feasibility of simultaneously maximizing all covariances(or other index of linkage strength), one for each possible dataset pairamong all datasets. Mathematically, the existence and uniqueness of thisnew object function's global maxima will be discussed.

Additional alternative inter-network analysis tool investigation: Inaddition to PLS, other methodologies will be explored for analyzingmulti-modality neuroimaging/non-imaging datasets. See Experimentaldesign & Methods section for more.

Significance:

Bio-mathematical methodology development: To our knowledge, our proposedmulti-modality inter-network analysis approach is the first of thiskind. It is our attempt to answer the call of analyzing multi-datasetsof unusual sizes simultaneously and in a systematic manner. Aside fromits relevancy to the biomedical especially neuroimaging studies,methodological questions raised in this endeavor are mathematicallychallenging. They will for sure initiate and stimulate necessarytheoretical discussions which in turn will provide insights on theproposed approach's application and further development. Though ourprimary focuses of the current proposal will not be mathematical theoremoriented, we will be just as rigorous in introducing various alternativeobject functions for MBPLS, in proposing related optimizationstrategies, and in defining small probability events in the calculationof the type-I errors and statistical powers. Furthermore, our logicallyconstructed Monte-Carlo simulations will be inspiring for the furtherpursue of mathematical theorem oriented discussions. In this study, wewill not only attempt to define these biomedically relevant challenges,but also actively initiate the communication with theoretically orientedmathematician/statisticians to advance the developments.

Neuroimaging multi-biological process analysis procedure: We believe ourinter-network multi-modality PLS is not a simple extension of theintra-modality PLS method. Rather it is novel in several facets. First,the inter-network PLS seeks direct linkage among images of differentmodalities. The linkage strengths and the singular images provideinformation complementary to that given by analysis of each imagedataset alone, univariate or multivariate. With this direct linkageapproach, different multi-physiological/metabolic processes andanatomical structural information can be investigated andcross-referenced. Moreover, this multiple process investigation can beperformed with or without in reference to experimental conditions orbehavior measurement (i.e., all under rest condition). Secondly, theproposed global index (or a set of global indices) combined with somepre-specified nodes on the singular image set as surrogate markers isinnovative together with the Monte-Carlo simulation for the statisticalpower and type-I error calculation (see Research Plan). Finally,computation strategies will be developed to make the proposedmulti-modality inter-network analysis procedure feasible.

Inter-network multi-modality analysis tool for Alzheimer disease study:With this inter-network analysis tool, the relationship between brainstructure and brain function, for example, can be investigated helpingus to evaluate differential genetic risks of AD associated with 0, 1 or2 copies of APOE-ε4 allele in our NIH sponsored on-going longitudinalneuroimaging study. Similarly, this tool can also be very helpful tounderstand the progression of AD disease, the conversion from mildcognitive impairment (MC) to AD in the other NIH-sponsored AlzheimerDisease Neuroimaging Initiative (ADNI) study (see D.1 data section formore).

Potential applications to other biomedical research/clinical areas: Webelieve that PLS is a tool not only for the imaging datasets, but alsofor others as well such as those from genomics or bioinformatics. Forexample, the linkage among the brain structure, brain function and thegenomic makeup can be characterized and explored with the use of MBPLS.Efforts will be made so the implementation of the algorithm as genericand applicable as to data not only from the neuroimaging studies butfrom multi-fields.

Finally, it is worth to note the need for us to consider the alternativeMBPLS object function for the study of the linkage among them withoutdesignating one dataset as dependent dataset and others as predictors.To study the PIB-FDG pattern in relation to FDG-PET and volumetric MRI,for example, one can certainly treat PIB-FDG dataset as dependentdatablock. However, a relationship among these three datasets with samelabeling could provide fair view of the data.

C. Preliminary Studies

C.1 Implementation of DBPLS for Voxel-Based Neuroimaging Data

C.1.1 the Iterative Way to Compute PLS: Power Algorithm

With the notation introduced earlier, it is obvious that the size of thesquare matrix Ω is the number of voxels within the brain volume(assuming the same number of voxels for both imaging datasets).

To make the computation possible, we partition each of the huge matrices(X, Y, Ω and other intermediate ones) into a series of small matriceswhich are only read in, one at a time, into the computer memory whenneeded. To make this strategy works, the only allowed matrix operationsare those that can act separately on sub-matrices and result insub-matrix form. One example of such operation is the multiplication ofX′ by Y. To use the strategy outlined above for the singular valuedecomposition (SVD) calculations related to DBPLS, we adopted theso-called power algorithm which is iterative in nature (Golub, GH andVan Loan, C F 1989). The operations involved at each of the iterationsare only matrix×vector, vector×matrix, and vector×scalar which are allseparable onto the sub-matrices.

The MATLAB code for SVD calculation using power algorithm in comparisonto MATLAB routine svds.m is given in the Appendix. Note both the examplepower algorithm code and svds.m need the whole matrix to be in memory.In implementing power algorithm in our PLS analysis, all the matrix byvector, vector by scalar multiplications are done by reading in onesub-matrix a time.

C.1.2 Efficient DBPLS Implementation Via Matrix Size Reduction

Assume the data matrix X is n by P_(X)(X_(n×P) _(X) ) with n<<Px andrank(X)≦n. Without losing generality, we assume rank(X)=n. The row spaceof X is with an orthornorm basis, e=(e₁ ^(T) e₂ ^(T) . . . e_(n)^(T))^(T) each as a row vector satisfying: e_(i)e_(j)={_(0, i≠j)^(1, i=j).

This basis, for example, can be the one via principal component analysison the matrix X. Note that there are infinite many such bases. X can beexpressed as X=X₁e where X₁ is a full-rank n×n matrix.

Similarly, Y=Y₁f with f=(f₁ ^(T) f₂ ^(T), . . . f_(n) ^(T))^(T) beingorthornorm basis of the space spanned by the rows of Y. Thus,X^(T)Y=e^(T)X₁ ^(T)Y₁f. On the other hand, SVD gives X^(T)Y=USV^(T) X₁^(T)Y₁=U₁S₁V₁ ^(T), where U, V, U₁ and V₁ are, in general, unitarymatrices. Thus, we have USV^(T)=e^(T)U₁S₁V₁ ^(T)f

Motivated by this derivations, we implemented the calculation of X₁, Y₁,and svd of X₁ ^(T)Y₁ (i.e., the calculation of U₁, S₁ and V₁). Then weused matrix e or f to transform the solutions back to the space of theoriginal matrices X and Y. Theoretically, however, we are not claimingthat the first n diagonal elements of S equal the n diagonal elements ofS₁ and we are not claiming that there exists an equal relationshipbetween the first n columns of U with the first n columns of e^(T)U₁ orbetween the first n rows of V and the first n rows of V₁ ^(T)f.

In any case, we will further explore these relationships described here(see research plan below) and seek the possibilities to take theadvantages of the efficient computing for the reduced matrices (asinitial value for the iterative power algorithm, for example).

C.2. Assessment of Statistical Significance and Reliabilities

C.2.1. Jacknife Procedure

Experimental design permitting, the leave-one-out procedure is aneconomic way to empirically validate our inter-network analysis strategyusing available data. The Jacknife cross-validation procedure could bean efficient way to demonstrate the latent variable pair as powerfuldiscriminators (in discriminate analysis in C.3) or indices oflongitudinal decline (in power analysis).

C.2.2. Bootstrap

Bootstrap resampling technique can be used to estimate the voxel-wisestandard errors of the singular images (for imaging data) or theelement-wise standard error of the vector w and c in general. Thesingular image can be scaled by voxel-wise standard error forstatistical significance assessment.

C.3. Preliminary Empirical Validation and Application

C.3.1 Subjects and Imaging data

To empirically validate the proposed DBPLS method for examining thefunctional/structural linkage between FDG-PET and MRI datasets in thispreliminary study, FDG-PET/MRI data from 15 young adults (31.3±4.8 yearsold) and 14 elder adults (70.7±3.5 years old) were used. All of them areparticipants of our on-going longitudinal study of Apolipoprotein ε4(APOE-ε4), a generic risk factor of Alzheimer disease, and all arenon-carriers of APOE-ε4 (i.e., they have 0 copies of APOE-ε4 alleles).Subjects agreed that they would not be given information about theirapolipoprotein E genotype, provided their informed consent, and werestudied under guidelines approved by human-subjects committees at GoodSamaritan Regional Medical Center (Phoenix, Ariz.) and the Mayo Clinic(Rochester, Minn.).

The subjects denied having impairment in memory or other cognitiveskills did not satisfy criteria for a current psychiatric disorder anddid not use centrally acting medications for at least two weeks beforetheir PET/MRI session. All had a normal neurological examination.Investigators who were unaware of the subjects' APOE-ε4 type obtaineddata from medical and family histories, a neurological examination, anda structured psychiatric interview. All of the subjects completed theFolstein modified Mini-Mental State Examination (MMSE) and the HamiltonDepression Rating Scale and all but one subject completed a battery ofneuropsychological tests.

PET was performed with the 951/31 ECAT scanner (Siemens, Knoxville,Tenn.), a 20-minute transmission scan, the intravenous injection of 10mCi of 18F-fluorodeoxyglucose, and a 60-min dynamic sequence of emissionscans as the subjects, who had fasted for at least 4 hours, lay quietlyin a darkened room with their eyes closed and directed forward. PETimages were reconstructed using the back projection with Hanning filterof 0.40 cycle per pixel and measured attenuation correction, resulting31 slices with in-plane resolution of about 8.5 mm, full width at halfmaximum (FWHM) and axial resolution of 5.0-7.1 mm FWHM, 3.375 slicethickness and 10.4 cm axial field of view. The rate of glucosemetabolism (milligrams per minute per 100 g of tissue) was calculatedwith the use of an image-derived input function, plasma glucose levels,and a graphic method (Chen, K. et al. 1998). Glucose metabolism in thewhole brain was calculated in each subject as the average measurementfrom all intracerebral voxels (including those of ventricles) inferiorto a horizontal slice through the mid-thalamus.

MRI data was acquired using a 1.5 T Signa system (General Electric,Milwaukee, Wis.) and T1 weighted, three-dimensional pulse sequence(radio-frequency-spoiled gradient recall acquisition in the steady state(SPGR), repetition time=33 msec, echo time=5 msec, α=300, number ofexcitations=1, field of view=24 cm, imaging matrix=256 by 192, slicethickness=1.5 mm, scan time=13:36 min). The MRI data set consisted of124 contiguous horizontal slices with in-plane voxel dimension of 0.94by 1.25 mm.

The example data set was analyzed by PLS having two group subjectspooled together (group membership information is not used in theanalysis). We also refer this group membership blind PLS analysis asagnostic PLS

C.3.2 Data Pre-Processing

Image pre-processing was performed using the computer package SPM99(http://www.fil.ion.ucl.ac.uk/spm, Wellcome Department of CognitiveNeurology, London). Improved procedure for optimal MRI segmentation andnormalization was used to discount the effect of non-brain tissue ingenerating gray tissue probability map for each subject on the MNItemplate space (created by Montreal Neurological Institute). Briefly,this optimal procedure first segments the MRI data on each subject'sbrain space, masks the segmented gray tissue map with careful reviewingthe mask first to eliminate any non-brain part. Then, the procedureestimates the deformation parameters comparing the masked gray mattermap to the one on the MNI template coordinate space, and subsequentlydeforms the raw MRI data which was then segmented to create the graymatter map on the MNI template space. Both modulated and un-modulatedgray matter maps were created. The gray tissue maps were also re-sampledto 26 slices (thickness of 4 mm), each slice is a matrix with 65 by 87voxels of 2 mm. Finally, a common mask was created containing only thosevoxels whose gray matter intensity values is 0.2 or higher on allsubjects. PET data was also deformed to the MNI template space with thesame voxel size and slice thickness. The same 20% common mask wasapplied to the PET data as well. Finally, PET and MRI data were smoothedrespectively to make their final resolutions compatible.

C.3.3 Preliminary Results

The PLS algorithm was implemented using MATLAB (MathWorks, MA) on an IBMA31 laptop running linux operating system.

First, the accuracy and reliability of the sub-matrix based Poweralgorithm was tested against the MATLAB SVD implementation (svd.m andsvds.m) using randomly generated matrix of varying sizes (100 by 100 upto 6500 by 6500). It was found that the implementation of poweralgorithm was equivalent to its MATLAB counterpart. However, for acomputer with 1 GB RAM and 1 GB swap space, MATLAB svds.m crashed for amatrix of a moderately large size (6500 by 6500), speaking for the needto divide huge matrices into smaller ones.

For the example MRI/PET datasets, each row of the matrix X was formed byarranging the voxels of one subject's brain into a row vector. Thus, thenumber of rows in matrix X is the number of subjects and the number ofcolumns is the number voxels in the brain mask.

When no attempt was made to first reduce the matrix size, the computingof the first singular image pair and the associated singular value tookabout 70 hours after some code optimization.

Not surprisingly, it was found that the PET-PLS subject scores and theMRI-PLS subject scores are closed correlated (R=0.84, p<7.17e-09). Moreinterestingly, as shown in FIG. 2A, there is a total separation betweenthe young and old subject group (open circles: old subjects; closedcircles: young subjects).

The first singular images for PET and MRI, as shown in FIG. 2B (leftpanel PET, right panel: MRI), were created with an arbitrary thresholdof p=0.05 to both positive and negative values of the singular imagesafter normalization by the bootstrap estimated standard error (p=0.05 isjustified as there is no need for multiple comparisons). Realizing thatthe signs of the singular images are relative and that theinterpretation of the PLS results is an unfamiliar territory, ourcurrent bio-physiological understanding of the dual-patterns is asfollows: The combined pattern indicated consistent lower gray matterconcentrations and lower cerebral metabolic rate for glucose (CMRgl)occurred concurrently in medial frontal, anterior cingulate, bilateralsuperior frontal and precuneus regions; posterior cingulate andbilateral inferior frontal regions were only seen (with negative patternweights) on the PET but not on the MRI; some white matter regions,caudate substructure, and occipital regions were shown to beconcurrently preserved or on the gray matter distribution along. Thesecross-sectional young and old group MRI/PET findings indicate that, invery healthy adults, age group difference is associated with regionallydistributed and interlinked dual network patterns of brain gray matterconcentration and CMRgl changes, as measured with MRI voxel-basedmorphometry and FDG-PET.

It is also worth to compare the PLS results against the SPM findings.SPM was performed contrasting the young and old subject groupsseparately for the PET dataset and the MRI gray matter dataset(voxel-based morphometry analysis). We found overall patternsimilarities between the PLS singular images and the SPM T-score maps aswell as multiple apparent focused differences. In contrast to SPM,however, inter-network PLS combines information from both modalities andprovides a global index (pair) for which can be used as a powerfuldiscriminator. For example, the multiple comparison corrected globalmaxima of the PET or MRI is significant at p=0.005 (corrected), thePET/MRI PLS latent variable is p<2.32e-18 contrasting young and oldsubjects without the need to correct multiple comparisons.

The matrix size reduction technique improved the computing speedsignificantly. In fact, the PLS took less than a minute to finish, withone time overhead effort (in couple of hours) to construct the orthonormbasis, e and f, respectively for X and Y. We found striking similaritiesPLS results with or without reducing the matrix sizes first. However,differences existed between these two approaches both in terms of thespatial patterns of the singular images and in terms of the latentvariable numerical values and empirical distributions.

The differences were also evident when we performed Jacknife analysis.Our purpose of the Jacknife analysis is to examine the accuracy ofclassifying the subject who was left out at each of 29 runs. A linearclassifier was determined first in each run based on the information ofthe remaining 28 subjects. The classification is to assign the left-outsubject to young or old group based on his/her PET and MRI latentvariable numerical values against the classifier. 100% accuracy wasobtained for the PLS procedure without the matrix size reductionperformed. With the matrix size reduction, 3 of 29 subjects weremisclassified (89.7% accuracy).

C4. Finding Summary of Preliminary Studies

The findings and their implications of our preliminary study can besummarized as:

-   -   (1) Iterative Power algorithm is numerically identical to MATLAB        svd routine    -   (2) Though computationally expensive, the PLS is still feasible        without matrix size reduced for research settings    -   (3) The combination of structural and functional imaging data        increased sensitivity    -   (4) The inter-network PLS results are in general consistent with        the univariate SPM findings but with more increased statistical        power    -   (5) PLS latent variables could be construction parts for global        index for detecting changes, for distinguishing group        differences, or for classifications,    -   (6) There exists potential possibility to improve the computing        speed    -   (7) Non-parametric statistical testing and validation procedure        are integrated parts of the PLS implementation.

D. Experimental Design & Methods

D.1: Data

No need for any new data to be acquired under this proposal. Our plan isto use data acquired under the supports of various existing grants orthat to be started. Our use of human subjects' data will be strictlyobedient by the HIPPA regulation and any requirements fromlocal/institutional IRB.

D.1.1 MRI, FDG-PET Data from Our NIH Sponsored Longitudinal APOE-ε4Study

With over more than 160 healthy subjects followed longitudinally (someof them have 5 or more visits already), this NIH sponsored project (NIMHMH057899-06), on which Dr. Chen and Dr. Alexander are listed asinvestigators, Dr. Reiman as PI and Dr. Caselli as co-PI, isunprecedented in many aspects. It will be our first choice of our PLSapplications especially with our specific aim of developing a predictionscheme and constructing a clinical diagnostic scheme based on crosssectional and longitudinal FDG-PET and MRI data (to a limited extent, asthe data are all from normal subjects. See D.1.3). Thus, both crosssectional and longitudinal datasets will be considered. Moreover, withour implementation of MBPLS and the availability of theneuropsychological (NP) data, PLS application to triple datasets (MRI,PET and NP data) will be on the top of our priority list. Theconventional MBPLS application will be aimed for the AD diagnosis,prediction of disease onset or conversion to MCI and treatmentevaluation. In addition, the MBPLS with the newly proposed objectfunction will be used to look for inter-linkage among the imaging andnon-imaging datasets.

The patient recruitment procedure, MRI/PET imaging data acquisitionprocedure, neuropsychological measurements, and the IRB regulations(requirement of consent form etc.) are the same or almost identical asthe ones described in the preliminary study section.

D.1.2 MRI, FDG-PET Data from Our Alzheimer Association Sponsored APOE-ε4Study

The preliminary study described in this grant application is actuallybased on the data from this Alzheimer Association sponsored study. Assuch, description of the data can be found in the ‘Preliminary Studies’section. Again, Dr. Chen and Dr. Alexander are listed as investigators,Dr. Reiman as PI and Dr. Caselli as co-PI.

D.1.3 MRI, FDG-PET Data of AD Patients, MCI Patients and HealthySubjects Under ADNI

The Alzheimer disease neuroimaging initiative (ADNI) is one of thelargest projects sponsored by NIH in its history. Dr. Chen, Dr.Alexander and Dr. Caselli are investigator and Dr. Reiman is co-PI onthis project which started in the early part of 2005. The PI is Dr.Michael Weiner of UCSF. As many as 800 subjects will be recruited fortheir participation over two-year interval. Longitudinal MRI data willbe obtained for all 800 subjects and half of them will be having FDG-PETas well. Since this project involves AD patients, MCI patients andnormal subjects, we will be able to evaluate the use of PLS tocharacterize the normal aging, the disease progress, and the conversionto MCI and to AD. More importantly, we have more opportunities ofdeveloping a prediction scheme and constructing a clinical diagnosticscheme based on cross sectional and longitudinal FDG-PET and MRI data.

D.1.5 PLS Analysis on Non-Imaging Data with or without Neuroimaging Data

We will actively explore the possibility of applying our multi-modalityinter-network PLS approach in and out the neuroimaging field. Microarraydata from genomics study will be made available through our connectionat the Translational Genomics Institute at Phoenix. The PLS applicationto genomic data will be with and without available neuroimaging data.The multi-dataset PLS analysis of genomic, neuroimaging data (FDG-PETand MRI) and neuropsychological measurements will be performed afteranalyzing, by Dr. Papassotiropoulos, blood samples from a subset of theparticipants of our longitudinal APOE-ε4 study to obtain their genomicinformation which are being planed and supported by other sources.

D. 1.6 Optimized Data Pre-Processing

In the preliminary findings section, we introduced some pre-processingsteps for the FDG-PET and MRI data. The pre-processing procedure will bestudied further especially with in mind that datasets of other typescould be part of the PLS analysis. The pre-processing steps that are ofcommon interest to many analyses, such as spatial normalization,smoothing, some issues related to voxel-based morphometry (VBM) etc.,will not be the focuses of the current investigation. (We are keenlyaware of the debates on VBM, and confident that the new developmentsimplemented in the new version of SPM5 will address that to asatisfaction. Advances on these areas will be followed closely andadopted in our pre-processing steps. Pre-processing steps that are morespecific to PLS (or multivariate analysis in general) will beinvestigated, and their effects evaluated, carefully. Datastandardization, for example, was traditionally performed by removingthe mean and unitizing the standard deviation. We will consider variousways of incorporating the whole brain measurement into thisstandardization, such as proportional scaling or analysis of variance(ANOVA). This conventional standardization will also be reviewed for thelongitudinal study for the use of baseline average vs. averages atfollowup times. Other pre-processing issues we will investigate includethe assumption of multiplicative modulation of the global on regionalmeasurements (like the one of SSM (Alexander, G. E and moeller, J 1994))for some or all datasets and the use of baseline data as a priori forfollowup gray tissue segmentation.

D.2 DBPLS and MBPLS Implementation and Validation

D.2.1 MBPLS Implementation

Our previous PLS implementation focused on only DBPLS. Extensive effortswill be made for voxel-based MBPLS. On a voxel-by-voxel basis, we willfirst attempt to have the well-established MBPLS algorithm programmedfor neuroimaging datasets also taking the presence of non-imagingdataset(s) into considerations. Subsequent validation, non-parametricstatistical procedure and its use for real data analysis will follow asdescribed elsewhere in this proposal.

In the methodological development session (see below), we propose toinvestigate the linkage among multi-datasets without designating one asdependent datablock and the rest as independent (predictor) datablocks.The methodological and theoretical investigation will be accompaniedwith its implementation first on personal desktop computers. In fact,the test code implementation and evaluation will be an important part ofthe methodological development. Once (and only after) its mathematicalappropriateness and feasibility are fully understood, efforts will bedevoted to make it available on the super compute system. Also,completion of the package will be marked as its flexibility of dealingwith voxel-based, ROI-based imaging data or non-imaging data in general(see below).

D.2.2 Voxel-Based and ROI Based Implementations

Our current implementation of DBPLS is voxel-based. Though no extraefforts are needed in the computing part for the ROI based data as longas the data are fed to the program in proper format, it is not a trivialtask to have a set of ROI chosen which are appropriate for brainfunctions in general, or designated only to specific brain diseases suchas AD. With our primary AD research interest, a list of brain regionsaffected by AD will be generated based on our own research (Alexander,G. E. et al. 2002) and others (Minoshima, S. et al. 1995; Ibanez, V. etal. 1998; Silverman, D. et al. 2001). These brain regions will becarefully delimitated on the high resolution MRI template in the MNIcoordinate space. The reliability of the ROI definition procedure willbe examined (intra- and inter-raters test-retest) if some of the ROIneed to be manually defined (for this purpose, we plan to use computerpackage MRIcro by Chris Rorden [www.mricro.com]). We also plan totransform these ROIs over to our customized template (for AD patients orfor healthy subjects) using automated template-based ROI generationprocedure (Hammers, A. et al. 2002). Published, widely used andwell-documented ROI procedure as well as the results (e.g., the up to200 ROIs that have been carefully defined by UCLA researchers in theirefforts to automate the clinical diagnosis of AD) will be activelysearched and utilized to minimize our own efforts.

D.2.3 PLS Validation

In the preliminary findings section, we reported the consistency of ourPET/MRI PLS findings with SPM analysis results of PET and MRIseparately. We plan to validate further our multi-modality inter-networkPLS approach in contrasting its results to the results of univariateanalysis for individual dataset (such as by SPM). The consistencybetween inter-modality PLS and intra-modality univariate analysisvalidated indirectly the PLS approach. More importantly, the increasedsensitivity by multi-dataset PLS, as found in our preliminary study, isdemonstrated the expected power. The contrast between PLS and theunivariate analysis, with the insights to biomedical andbiophysiological processes, will also be helpful in understanding andinterpreting the PLS results.

Another important aspect of PLS validation is the reproducibility of theuncovered inter-network patterns (singular images) and the latentvariable pairs. Though Jacknife leave-one-out procedure is a soundcross-validation in this regard, repeating the analysis of the samebiological nature on imaging data acquired from a different group ofsubjects would be more assuring. The reproducibility study of this kindwill be performed for various studies including the young/old subjectstudy reported in the preliminary findings section as data from moresubjects are being acquired with the support of our longitudinal APOE-ε4project and others. (Note the number of subjects in each group preventedus from doing so in our preliminary study). Whenever permitting,subjects will be divided for two identical analyses for validatingreproducibility. Exactly like establishing an index for monitoringdisease progress or diagnosing disease onset, the group split will berepeated to the maximum number possible to increase the validationefficiency in terms of the use of the data available and programmingefforts will be made so the validation can be updated when new data areadded to our database.

D.3 Establishment of MBPLS/DMPLS as an Integrated Surrogate Marker forTreatment Evaluation and Disease Progress for AD

We will devote a significant effort in developing the multi-modalityinter-network PLS as a scheme which can be used in assessinglongitudinal changes with or without intervention and in describingdisease progressing especially for AD. For treatment evaluation, it isnow well recognized that the use of neuroimaging surrogate marker isassociated with much increased statistical power, reduced cost, andshortened study duration. More importantly, neuroimaging techniqueallows the treatment/prevention effects to be observed at the earlystage of the interested disease or even before its onset as demonstratedin our APOE-ε4 study. For AD disease progress or the brain alternationsbefore clinical symptoms (Reiman, E. M. et al. 1996; Reiman, E. M. etal. 2001; Reiman, Chen, Alexander, Caselli, Bandy, Osborne, Saunders,and Hardy 2004), it is now more and more common to acquiremulti-modality imaging and non-imaging data. On the other hand, therichness of the neuroimaging data has not been used optimally. The lackof full use of neuroimaging data is reflected by the fact thatunivariate statistics is the dominant analytical tool for almost allneuroimaging studies evaluating the effects of a treatment or diseaseprogress. In other words, a number of selected brain regions or a globalindex is often used for statistical power calculation, for diseaseprogress monitoring and for clinical diagnosis (often without correctingmultiple comparisons).

As a complement to the univariate approach, we propose and attempt toestablish intra- and inter-modality multivariate indices as ananalytical tool in studying treatment effects, in monitoring diseaseprogress, and potentially in diagnosing AD disease (usingcross-sectional as well as longitudinal data). The proposed approachwill enable a researcher to use the richness of the neuroimaging data tothe fullest. Consequently, increased statistical powers, reduced type-Ierrors and improved sensitivity and specificity are expected. On theother hand, the approach should not be too complicated andcounter-intuitive.

We propose to investigate the inter-network PLS feasibility as asurrogate marker following the procedures described below. Note that thebasic idea discussed here are applicable to both single-modality andmulti-modality datasets.

D.3.1 Longitudinal PLS Analysis

As our preliminary findings were cross-sectional, we will brieflydescribe several approaches of dealing with longitudinal data here. A)If the longitudinal data are only for two time points (baseline andfollowup), then the subtraction image could be created and entered intothe inter-network PLS analysis after taking care of variation in thetime intervals and in the whole brain measurements; B) data at differenttime points can be treated as separated datablocks and enter them allinto the analysis directly. In doing this, we will need to investigatemeans to have the longitudinal information incorporated; C) PLS can beperformed separately for data from each time point followed by theexaminations of changes in latent variable and in singular images usingconventional statistical tools; D) Results from univariate analysis(such as SPM) can be the starting point of further PLS analysis. Forexample, longitudinal voxel-wise regression coefficients (the slope,e.g.,) can be subjected to further PLS analysis (cross patient groups,e.g.). We will focus on A) and D) first in our proposed study. Note thenext subsection is with in mind the discussion of this part.

D.3.2 Index Establishment.

An index, or a set of indices, is a measure of longitudinal changes withor without treatments. In the simple univariate index case, the CMRgldecline (the difference between the baseline and followup scans) for agiven brain location is such an ideal index. The effects of an evaluatedtreatment are reflected as a measurable reduction of the decline. Thedecline without the treatment and the decline reduction with treatmenttogether with their variation are usually the starting point for thedetermination of the number of subjects needed in a new trial with adesired statistical power.

First, potential candidates for such as an index or a set of indicescould be the latent variable pair(s) following the same logic thinkingin the well established statistical power procedure, but also taking theinter-voxel covariance and inter-network covariance into considerations.With the directionality of the latent variable made consistent withlongitudinal decline (sign of the latent variable and the weights isrelative, and will not affect the linkage assessment), for example, thefirst latent variable pair can be combined to form a single index or canbe used as a bivariate indices to enter into power calculation (note themaximum covariance does not imply maximal correlation). Since the latentvariables themselves summarize both intra-modality and inter-modalitylinkages, the power calculated is not based on selecting a few voxel/ROIlocations and ignoring the relationship among them and with the rest ofthe brain. The same idea can be applied to the use of up to 2nd, 3rd ormore latent variable pairs, or an optimal combination of them, which isoptimally pre-determined in correlating to clinical outcomes, forexample.

Secondly, the singular-image within each dataset (the weights w in thefirst dataset, for example) can be used to construct indices forsubsequent statistical power or disease progress analysis. This ispossible because of the availability of the bootstrap estimated variancefor each weight (at each voxel). Again, with the establishment of theweight directionality (positive or negative) consistent with theunivariate voxel-wise CMRgl decline, a collection of the singular-imagevoxels where the weight are of significance (p<0.005, e.g.,) can bechosen. Note the selection of the voxel can also be guided with theresults of the voxel-based analysis.

Third and finally, the directional singular-image differences between ADpatients and MCI patient (AD research as our major application area ofthe current methodology proposal) can be utilized together with theBootstrap approach estimated voxel-wise variances. Like the individualvoxel CMRgl decline and the decline reduction in univariate poweranalysis for a single-modality PET study, the pattern/networkdifferences and their hypothesized reductions (either universal or brainregion dependent) could be foundation for determining the number ofsubjects for desired powers or the powers for given number of subjectsfor a multi-modality study (dual FDG-PET and MRI, for example) or basisfor reporting disease progress/severity.

D.3.3 Power Calculations

Power analysis can be performed for each of the latent variable pairsseparately and followed by the combined power (defined as theprobability observing at least one of these effects). This combinationprocedure is partially justified as the latent variable pairs areuncorrelated and are assumed Gaussian, therefore independent. For thechosen voxels over the singular-image (selected significant node-pointsover the spatial pattern) or itself, we propose to use Monte-Carlosimulation procedure for the type-I error and statistical powercalculation since no available software, to our knowledge, exists forsuch purpose (see B, Appendix for our own preliminary work presented inthe annual nuclear medicine meeting, 2004). Using the first latentvariable pair in a dual-dataset study as an example, our currentprimitive thinking of the simulation procedure is provided below.

The Monte-Carlo simulation package is based on the computer packageSPM99. The simulation starts with a 3D brain mask (provided by theresearcher) over a standard or customized brain space (e.g., MNItemplate space). Thus, spatial normalization, image alignments, etc.processings are not part of the simulation process. For each of Niterations (N=10000, for example), this Monte-Carlo simulation procedureconsists of the following steps: (1) For the type-I error calculation, a3D brain image of either of the two modalities for each of M subjects isgenerated as Gaussian random numbers on a voxel-by-voxel basis. Theimage is then smoothed according to the final image resolution of theanalysis. For statistical power calculation, the map generationprocedure is identical as above but with the averaged images of theapproximations by up to nth eigen-images for all M subjects (linearregression approximation, similar to approximation of the original imageby the first several PCA components). The voxel-wise variance andinter-voxel covariance estimated by Bootstrap will be incorporated asthe following. Assume E is the covariance matrix within the eigen-imagevoxels that are significant. This huge matrix is rank deficient due tothe fact that the number of voxels (variables) is most likely far morethan the number of observations (the number of Bootstrap resamplings).Thus, square matrix Ω exists such that Σ=Q′ΛQ, where Q′Q=QQ′=I and Λ isdiagonal with only the first rank(Σ) non-zero elements. Thus, one canquickly generate random vector x of length rank(Σ) with mean zero andcovariance matrix Λ, and random y=Q′x+a will have covariance matrix Σ,where a is the voxel-by-voxel mean. For power calculation purpose, thecovariance matrix can be replaced by correlation matrix so the final yis of unit variance and the mean of

$\frac{a}{\sqrt{{diag}(\Sigma)}}$

(effect size). In addition to dealing with the inter-voxel correlations,it is important also to note that the added effects assume that theGaussian variable is with unit standard deviation. The smoothingprocess, however, reduced the standard deviation to sub-unity levels.Thus, the original known effect sizes, relative to the smoothed randomfield, are much larger. Consequently, the statistical power could besignificantly over-estimated. To correct the over-estimation of thepower, each smoothed Gaussian random field is scaled by its newcross-voxel standard deviation priori to the introduction of thenon-zero effect sizes.

(2) The threshold of a given type-I error (5%, e.g.,) can be assessed by(2D) histogram constructed over the N simulations/realizations (2Dcorresponds to two imaging datasets). The type I error (the significancelevel) is estimated as the ratio of n over N, where n is the occurrencesof the hypothesized event (without effect of interest introduced). Amongseveral potential alternatives, the threshold of the type-I error, T,can be calculated over the 2D histogram as prob(√{square root over(x²+y²)}≦T)=1−α where α is the type-I error and x and y are the firstlatent variable pair. T is then used for the power calculation.Apparently, the closeness of singular-image based on the simulated Msubjects' data to the true one (or the one from analyzing the real data)should be examined as a part of this study (see reproducibility part inthis Research Plan). Other alternatives exist and will be probedfurther.

D.3.4 Inter-Network PLS Based Disease Progress and Clinical Diagnosis

Independent to the power analysis, the use of the inter-network PLS forexamining disease progress and clinical diagnosis should be based on andconfirmed with well-established criterion historically and on on-goingbasis. To illustrate, we will use our NIH sponsored longitudinal APOE-ε4study as an example. With more and more healthy subjects in ourlongitudinal APOE-ε4 study converted to AD or MCI, our first attemptwill be to establish such criterion using MBPLS with the conversion rateas the dependent block and FDG-PET, MRI as independent blocks. Thiscriterion establishment will be based on a subset of the subjects. Therest will be used for validation purpose. To increase the validationefficiency in terms of the use of the data available, the group splitwill be repeated to the maximum number possible. Programming effortswill be made so the validation can be updated when new data are added toour database. This procedure also lays the foundation for the use ofMBPLS as a predictor on the onset of disease.

D.3.5 Relative Importance of a Datablock in Terms of Statistical Powerand in Terms of Clinical Diagnosis

We will use the normalized block score as a measure of the datablockimportance. Though it is not methodologically challenging, the relativeimportance in contributing to the diagnosis is of significance of bothbiologically and financially. New indices of datablock importance willalso be looked into proper to the research questions raised.

D.4 Methodological Developments

D.4.1 New Object Function for MBPLS

The calculation of conventional MBPLS is based on distinguishing thedatasets as a single dependent dataset and one or more independent(predictor) dataset(s). This setting is ideal if the focus is to predictthe performance of the dependent block from the independent blocks (suchas for disease progress and clinical diagnosis). However, when there isno clear dependent-independent distinction among the datasets (FDG-PET,PIB-PET and structural MRI from a group of AD patients, e.g.,), or whenone's primary interest is to seek the inter-relationship among alldatasets, a new approach is needed. There are numerous intuitive ways tosetup criterion in terms of defining the inter-dataset covariance. Thechallenge is to find the ones that are mathematically and logicallyjustified, scientifically meaningful and computationally feasible. Wewill list a few such criterions here to motivate ourselves and others.In the followings, assume there are c datasets, X₁, X₂, . . . , X_(c).t_(k) is a latent variable representing X_(k) (k=1, 2, . . . , c),t_(k)=Σw_(i) ^((k))x_(i) ^((k)) where x_(i) ^((k)) is the ith column ofmatrix X_(k) and w_(i) ^((k)) is the corresponding weights (of unitnorm). The following object functions can be defined for the calculationof the latent variables: A) max(min_(k<l)(cov(t_(k),t_(l)), B)

${\max\left( {\prod\limits_{k < l}\; {{cov}\left( {t_{k},t_{l}} \right)}} \right)}.$

Notice that the covariance used in these expressions is unconditional(the effects of other datasets are ignored when calculatingcov(t_(k),t_(l))). More complicated schemes will be needed for theobject function which uses the conditional covariance instead. We willneed to investigate the existence, uniqueness, convergence, and speed ofconvergence of the solution for the optimization procedure using theabove defined object functions or others. Moreover, proper iterativeprocedure needs to be established for uncovering the second, third, etc.latent variable sets taking care of the effects of previous latentvariable sets and orthogonality issue. Section C of the Appendixprovides some preliminary results on our alternative MBPLS investigationeffort. We demonstrated the uniqueness and existence of the alternativeMBPLS solution with some additional constrains.

D.4.2 DBPLS Calculation with and without Matrix Size Reduction

Our previous results given earlier showed DBPLS results differences andsimilarities between directly calculating the latent variable usingiterative Power scheme and reducing the matrix size to their ranksfirst. Further theoretical examinations and theory-guided computersimulations are needed to unveil the causes of thedifferences/similarities and to develop procedures, when feasible, toaccount for the differences. The improved computational speed associatedwith the reduction of the matrix size is important for the proposednon-parametric statistical resampling procedures as they in general areiterative in nature. The bootstrap procedure can be performed inconjunction with the matrix size reduction technique to estimate thestandard deviation of the weight at each voxel location for each imagemodality. The robustness of the estimated standard deviation in regardsto this dataset size reduction technique will be investigated.

D.4.3 Explore Alternatives in Addition to PLS Approach or Based on PLSResults

Though the primary focus of this proposal is to establish DBPLS andMBPLS as a tool for the study of inter-network linkage and a way tocombine information from multi-datasets, we realize that there are otherapproaches to describe various aspects of the relationship amongmulti-datasets and to maximize the power combining information from eachdataset. We view the establishment of MBPLS and DBPLS as 1) one of manytools that will be used to investigate the multi-datasets systematically(i.e., as inter-network approach), and 2) an explorative tool forfurther applying other methodologies either data-driven, model-based, orhypothesis driven. These methodologies are well established forintra-modality single dataset study with a track record of successfulapplications. However, they may need to be further generalized forinter-modality, multi-dataset studies. The two methodologies we areinterested for such generalization are (inter-datasets) independentcomponent analysis (ICA), and (dual-dataset) structural equationmodeling (SEM). At this very early stage, our description here will beonly sketchy and conceptual in the context of our future researchdirection.

Multi-datasets ICA: We will only illustrate the concepts for thedual-dataset case. For the conventional single-dataset, one way toobtain the ICA solution is the minimization of mutual information(Hyvarinen, A. et al. 2001). With the same notations as above, the firstinter-dataset independent pair t and u is obtained by minimizing themutual information between t and u:

min{H(t)+H(u)−H([tu])}

where H(x)=∫p(x)log p(x)dx is the entropy for continuous randomvariable/vector x, and p(x) is the probability density function (pdf) ofx. Integration will be replaced by summation for discrete randomvariable/vector. Note this is not a full procedure by which allindependent component pairs are obtained. A conceptually intuitivenumerical approach for putting constrains on the mixing matrix ingenerating the dual-dataset ICA solution is being investigated by ourgroup. Other alternatives are also being investigated but will not bediscussed here.

Multi-Datasets SEM:

Results from either voxel-based PLS or ROI-based PLS will provideresearchers the covarying pattern within each imaging modality datasetand the linkage among these datasets. These pattern and linkageinformation can be further understood with the construction of a properquantitative (mathematical) model such as SEM. The generalization of thewell-known SEM to the case of inter-datasets seems natural andstraightforward at first. However, one needs to find a way todistinguish and summarize the link strengths between nodes within onedatasets and those across multi-datasets.

We again emphasize our current research focus is the inter-dataset PLS.The discussion of these additional techniques (inter-network ICA andinter-network SEM) will serve us as reminder that the development ofinter-network PLS is only a start of methodological investigation of themulti-dataset analytical strategy.

D.5 Feasibility Testing of the Proposed Methodologies

Understandably, the proposed procedure is relatively expensive incomparison to the univariate analysis and to the intra-modalitymultivariate network analysis. However, our previous findings suggestedit is feasible computationally to perform DBPLS on dual-imaging dataseteven without reducing the matrix size first as an analysis procedure forbasic research settings. It is also important to know that number ofsubjects will only affect the computation time marginally with thecalculation of the covariance matrix at the very beginning and thesubject scores at the very end. Thus, the reported computation time inour previous findings is of representative for a wide range of numbersof subjects/scans. In the context of computational feasibility, it isworth to note that conventional MBPLS as a clinical diagnostic tool or amarker for treatment evaluation is computationally efficient as itsdependent datablock is with single or limited number variable(s). Thusthe size of the covariance matrix is not a concern.

We are not satisfied at all with the current computation speed. As canbe seen throughout this Research Plan, a major effort is to efficientlyimplement the algorithm. Computational feasibility testing will be anintegrated part in each and every step of the implementation. Like inour preliminary study, this feasibility testing includes the followingthree parts: (1) algorithm is correct, mathematically sound. Theimplemented algorithm will be examined carefully against mathematicalderivations and compared to well-established computer package that canhandle only non-imaging data with sizes that are much smaller than thatof neuroimaging datasets. The comparison will use computer simulateddatasets of moderate sizes; (2) algorithm is computationally efficient.Each part of the algorithm will be optimized (vectorize all possibleoperation in MATLAB, e.g.). That optimization will be tested against thedataset with expected sizes in real study (for example, number ofvoxels). The data sets can be either from real study or via computersimulation; (3) the output of the algorithm is scientificallyinterpretable. This is exactly the same as D.2.4 PLS validation. Seethat part for details.

The three-step feasibility testing outlined above will be at each stepof each algorithm planned for investigation. This is especially true forthe newly proposed methodology development (such as the alternativeMBPLS object function described in D.4.1).

Other Examples

FIG. 7 is a graph that depicts a comparison between an exact agnosticPLS operation and the inexact agnostic PLS operation. The graph setsforth eigenvectors of a 29 by 200 random matrix. For this eigenvector,the x-axis is the exact agnostic PLS operation and the y-axis is theinexact agnostic PLS operation.

Spatial patterns are shown that were uncovered by inter-modality exactagnostic PLS operations seen in FIG. 2B, by the inexact agnostic PLSoperation seen in FIG. 6A, by the standard univariate SPM procedure seenin FIG. 8A, and by the voxel-by-voxel multivariate MRI-PET SPM seen inFIG. 8B. Other than the voxel-wise multivariate SPM seen in FIG. 8B, MRIand PET findings are displayed on the left and on the rightrespectively.

Motivated by the availability of the multi-neuroimaging datasets andencouraged especially by the success of the single-modality PLSapproach, we propose to extend the use of PLS for analyzing dual-imagingdatasets. We hypothesize that this inter-modality PLS can seek for themaximal and direct linkage among multi-datasets or optimally combineinformation from them for increased statistical powers.

More specifically, 1), we explore the use of PLS both agnostically andnon-agnostically strategies to analyze dual-modality neuroimaging data.Agnostic PLS is to seek direct linkage between two image-datasetsblinded with any subject group membership or scan conditions and toperform subsequent analysis relevant to the condition/group differences.Non-agnostic PLS, on the other hand, is to consider the group/conditiondifferences directly in combining the information from dual-imagingdatasets. 2), we propose a computationally feasible approach for theagnostic PLS to overcome the difficulty associated with the huge size ofthe covariance matrix between two neuroimaging datasets. 3), we putforward an implementation alternative to first reduce the covariancematrix size to improve the computational speed of the agnostic PLS. 4),we will lay out the framework of performing non-parametric inference orcross-validation procedures respectively for agnostic and non-agnosticPLS. Finally, 5), we empirically validate this inter-network PLSapproach by applying it to dual MRI/PET datasets from well separatedyoung and old healthy subject group and contrasting the findings in thecontext of the univariate SPM analysis, and intra-modality PLS approach(i.e., using only one of the two imaging dataset to seek thegroup/condition differences).

Methods

Subjects and Data

To empirically validate PLS for examining the functional/structurallinkage between FDG-PET and MRI datasets, FDG-PET/MRI data from 15 youngadults (31.3±4.8 years old) and 14 elder adults (70.7±3.5 years old)were used. All of them are participants of the ‘PET, APOE and aging inthe Preclinical Course of AD’ study supported by the Alzheimer'sAssociation. All are APOE-ε4 non-carriers. Subjects agreed that theywould not be given information about their apolipoprotein E genotype,provided their informed consent, and were studied under guidelinesapproved by human-subjects committees at Banner Good Samaritan RegionalMedical Center (Phoenix, Ariz.) and the Mayo Clinic (Scottsdale, Ariz.).

The subjects denied having impairment in memory or other cognitiveskills, did not satisfy criteria for a current psychiatric disorder, anddid not use centrally acting medications for at least six weeks beforetheir PET/MRI session. All had a normal neurological examination.Investigators who were unaware of the subjects' APOE-ε4 type obtaineddata from medical and family histories, a neurological examination, anda structured psychiatric interview. All of the subjects completed theFolstein modified Mini-Mental State Examination (MMSE) and the HamiltonDepression Rating Scale and all but one subject completed a battery ofneuropsychological tests.

PET was performed with the 951/31 ECAT scanner (Siemens, Knoxville,Tenn.), a 20-minute transmission scan, the intravenous injection of 10mCi of ¹⁸F-fluorodeoxyglucose, and a 60-min dynamic sequence of emissionscans as the subjects, who had fasted for at least 4 hours, lay quietlyin a darkened room with their eyes closed and directed forward. PETimages were reconstructed using the back projection with Hanning filterof 0.40 cycle per pixel and measured attenuation correction, resultingin 31 slices with in-plane resolution of about 8.5 mm, full width athalf maximum (FWHM) and axial resolution of 5.0-7.1 mm FWHM, 3.375 slicethickness and 10.4 cm axial field of view. The rate of glucosemetabolism was calculated with the use of an image-derived inputfunction, plasma glucose levels, and a graphic method (Chen et al.1998). Glucose metabolism in the whole brain was calculated in eachsubject as the average measurement from all intracerebral voxels(including those of ventricles) inferior to a horizontal slice throughthe mid-thalamus.

MRI data was acquired using a 1.5 T Signa system (General Electric,Milwaukee, Wis.) and T1 weighted, three-dimensional pulse sequence(radio-frequency-spoiled gradient recall acquisition in the steady state(SPGR), repetition time=33 msec, echo time=5 msec, α=30°, number ofexcitations=1, field of view=24 cm, imaging matrix=256 by 192, slicethickness=1.5 mm, scan time=13:36 min). The MRI data set consisted of124 contiguous horizontal slices with in-plane voxel dimension of 0.94by 1.25 mm.

Data Pre-Processing

SPM99, a software package designed for the analysis of brain imagingdata sequences, was used for image pre-processing. The optimal MRIsegmentation and normalization procedure (Good et al. 2001) was used todiscount the effect of non-brain tissue in generating a gray tissueprobability map for each subject on the MNI template space (created byMontreal Neurological Institute). Both modulated and un-modulated graymatter maps were created. The gray tissue maps were also re-sampled to26 slices (thickness of 4 mm), each slice a matrix with 65 by 87 voxelsof 2 mm. Finally, a common mask was created containing only those voxelswhose gray matter intensity values is 0.2 or higher on all subjects. PETdata was also deformed to the MNI template space with the same voxelsize and slice thickness. The same brain mask was applied to the PETdata as well. Finally, PET and MRI data were smoothed respectively tomake their final resolutions compatible (final full width at halfmaximum is 15 mm for both smoothed MRI and PET).

PLS with Deflation

We adopted the one that deflates data matrices by projecting informationonto the previous latent variable pairs. This guarantees theorthogonality of the extracted component in all data spaces (Hoegaertset al. 2003). The PLS procedure with deflation scheme is describedbelow.

PLS uncovers the maximal covariance among a pair of latent variables,linearly constructed respectively from each of the two datasets.Starting from original data matrices X and Y (with standardizationnecessary), the first latent variable pair is constructed as follows:The latent variable of X is t=Σw_(i)x_(i) where w_(i) is a scalar forrandom variable x_(i) which is the i^(th) column of X (i=1, 2, . . . ).In matrix form, t=Xw where w=(w₁, w₂, . . . )^(T) with ∥w∥=1. Forimaging dataset, index i refers to the i^(th) voxel in the brain volume.Similarly, the Y latent variable can be expressed as u=Yc (∥c∥=1).Again, we refer to t and u as the first latent variable pair. In thecontext of agnostic PLS, we refer to w and c as (the first) singularimage of X and Y respectively as w and c can be mapped back to imagespace and displayed. The covariance of the two latent variables, t andu, is therefore cov(t,u)=w′X′Yc (assuming zero mean). The maximalcovariance value with respect to w and c can be proven to be the squareroot of the largest eigenvalue of the matrix Ω=[X′YY′X] with w being thecorresponding eigenvector of Q, and c being the correspondingeigenvector of Y′XX′Y The second latent variable pair can be constructedin a similar way after the contributions of the first latent variableare regressed out (deflated) from X and Y as follow: Express

${p_{1} = \frac{X^{\prime}t}{{t}^{2}}},{q_{1} = \frac{Y^{\prime}u}{{u}^{2}}},{r_{1} = \frac{Y^{\prime}t}{{t}^{2}}}$

and calculate new X₁ and Y₁ as X₁=X−tp₁′, Y₁=Y−tr₁′. The same procedurewill then be repeated for the new X and Y₁ matrix pair to construct thesecond latent variable pair. The third and remaining latent variablepairs will be calculated similarly (up to the L^(th) pair, whereL=rank(X)). Note that the deflation scheme described here is areflection of the fact that Y is designated as the dependent datablockand X as the independent datablock.

Multi-Block PLS

The PLS introduced so far is referred to as dual-PLS (DPLS for dualdatasets) and is for our agnostic PLS especially when both X and Y areimaging data. When one is interested in the relationship between adependent block, Y and more than one independent block, X₁, X₂, . . . ,X_(m), multi-block PLS (MPLS) is needed. As will be seen, ournon-agnostic PLS is actually MPLS in nature. The main difference betweenthe DPLS and MPLS appears when one attempts to uncover the latentvariables 2 and up. To start, MPLS uncovers the first latent variablebetween Y and X=[X₁ X₂ . . . X_(m)] exactly the same way as DPLS. TheDPLS deflation step described above, however, will mix contributionsfrom various X blocks and makes the result interpretation difficult.Various deflation schemes were proposed. Following the suggestion by(Westerhuis and Smilde 2001), we only deflate Y-block while keepingX-blocks untouched.

Agnostic PLS Versus Non-Agnostic PLS

In performing non-agnostic PLS, the young and old group membership isthe matrix Y as our main interest is the difference between these twogroups. In this case, Y is actually a column vector with value 1 and 2for young and old subject respectively. Alternatively, one could alsoform Y with individual subjects' age. The X block is formed by poolingPET and MRI data together, X=[PET MRI], where PET is n×P_(x) data matrixformed from the PET-FDG data. n is equal to the number of subjects andP_(x) is the number of brain voxels over the brain mask. The data matrixMRI is defined similarly. We referred this as non-agnostic PLS as thisMPLS procedure directly uses group membership as the dependent block.Apparently, type-I error assessment coming out directly from thisnon-agnostic PLS cannot be trusted, and additional procedure is neededto seek the true type-I error (see below).

The agnostic PLS, in contrast, seeks the directly linkage between thedual image datasets, PET and MRI, without referring to the old/youngsubject differences. Should the difference between the old and youngsubject be the primary source of variation, the agnostic PLS uncoveredmaximal covariance will have the power to distinguish the two groupsnaturally and without too much concern about the type-I error associatedwith the examination of the group differences (but see Bootstrap andjacknife procedures below).

Agnostic PLS Implementation Via Iterative Power Algorithm: Exact Method

It is obvious that the size of the square matrix Ω is the number ofvoxels within the brain volume (assuming the same number of voxels forboth imaging datasets). To make the computation possible, we partitionedmatrix Ω and others into a series of small matrices which are only readin, one at a time, into the computer memory when needed. To make thisstrategy work, the only allowed matrix operations are those that can acton sub-matrices and result in sub-matrix form. An operation of this kindis the matrix multiplication, for example. To use the strategy for thesingular value decomposition (SVD) related to PLS calculation, weadopted the so-called power algorithm which is iterative in nature(Golub and Van Loan 1989) (see appendix A for an illustrative piece ofMATLAB codes). The operations involved at each of the iterations areonly matrix x vector, vector x matrix, and vector x scalar which are allseparable onto the sub-matrices.

Agnostic PLS Implementation Via Matrix Size Reduction: Inexact Method

Assume the data matrix X is n by P_(X)(X_(n×P) _(X) ) with n<<P_(x) andrank(X)≦n. Without losing generality, we assume rank(X)=n. The row spaceof X is with an orthonormal basis, e=(e₁ ^(T) e₂ ^(T) . . . e_(n)^(T))^(T) satisfying e_(i)e_(j)={_(0, i≠j) ^(1, i=j). This basis, forexample, can be the one via PCA on the matrix X (Note that there are aninfinite number of such bases). X can be expressed as X=X₁e where X₁ isa full-rank n×n matrix. Similarly, we have Y=Y₁f. Thus,

X ^(T) Y=e ^(T) X ₁ ^(T) Y ₁ f  (1)

The singular value decomposition (SVD) of X^(T)Y=USV^(T) and X₁^(T)Y₁=U₁S₁V₁ ^(T). In these expressions, the U and U₁ are column-wiseorthogonal, and V and V₁ are orthogonal matrices. Equation (2) above canbe re-written as

USV ^(T) =e ^(T) U ₁ S ₁ V ₁ ^(T) f  (2)

Motivated by (2), we perform SVD on X₁ ^(T)Y₁ (i.e., the calculation ofU₁, S₁ and V₁) instead of X^(T)Y. We then used matrix e and f totransform the solutions back to the space of the original matrices X andY.

Though this approach is obviously inexact, our interest is to examinehow its results are compared to the exact approach (we are onlyinterested in the first n non-zero singular values and the associatedcolumns of U and V anyway).

PET-MRI Linkage Indices Related to Agnostic PLS

Since the agnostic PLS seeks directly the covarying patterns between thedual-image datasets, various scalar and images indices can be defined toexamine the relationship between the structural MRI and functional PET.In this current writing, two of them will be introduced and used. Thefirst is the squared correlation coefficient of the latent variable pairbetween the MRI dataset and the PET dataset. The second one is animage-wise index, referred to as the explanatory power. It is acorrelation coefficient map one over the PET image space. For each PETvoxel, the corresponding correlation coefficient is between the singlelatent variable of the X-block (MRI) and the PET measurement from thisvoxel.

Non-Parametric Statistical Procedure for Non-Agnostic PLS: Permutation

To assess the type-I error in testing the difference between the old andyoung subjects, 10000 row-wise random permutations were performed onmatrix Y and the MPLS procedure was run for each of this permuted Yblock and the unchanged X blocks. The histogram of unpaired t-testp-values over the 10000 runs is used to assess the type-I error. Onlypermutations that switch old/young group membership are counted towardthe total number of permutations performed.

Non-Parametric Statistical Procedure for Agnostic PLS Analysis: Jacknifeand Bootstrap

Jacknife Cross-Validation:

Agnostic PLS analysis (both exact and inexact) was repeated 29 timestaking one subject out each time. The resultant latent variable pairfrom the 28 subjects was used to construct a linear discriminator whichwas then applied to decide old/young group membership for the left-outsubject. This procedure allowed us to assess the classificationaccuracy.

Bootstrap Assessment of Statistical Significance:

With all 29 subjects included for the agnostic PLS, Bootstrap resamplingprocedure was run 100 times to estimate the voxel-wise standarddeviation of the singular images. The Bootstrap estimated standarddeviation was then used to scale the singular image pair for statisticalsignificance assessment.

PLS in Comparison with SPM

The PET and MRI dataset was each analyzed separately by univariate SPMcontrasting the old and young subjects (for the MRI data, the analysisis essentially the optimized voxel-based morphometry). To be consistentwith the PLS analysis, the global CMRgl and the total intracranialvolume (TIV) was accounted for by proportional scaling and the mask forthe PLS analysis (see pre-processing) was used in the SPM procedure. Inaddition to the univariate SPM, MRI gray matter maps and FDG-PET datawere run under the SPM multivariate mode with F-statistics. i.e., PETand MRI measurements were treated as bi-variates at each voxel. In bothSPM analyses, MRI and PET data from global maximal location(s) wereextracted and used jointly to test the young and old group differences.

The dual PET/MRI datasets were analyzed by non-agnostic PLS and byagnostic PLS with or without first reducing the sizes of the matrices ofX and Y. Again, the agnostic PLS analysis was conducted having the twogroup subjects pooled together (i.e., group membership information wasnot used in the PLS procedure. see Discussion section for more of ourrational of practice of this kind). The latent variable pair of theagnostic PLS was used jointly to test the young and old groupdifferences.

The spatial covarying patterns from the 5 analyses, univariate SPM,multi-variate SPM, non-agnostic PLS, agnostic PLS without matrix sizereduction, and agnostic PLS with matrix size reduction, were compared.The differences and similarities in the uncovered spatial patterns werevisually inspected. The spatial pattern is the single F-score map forthe multivariate SPM and the two t-score maps (one for FDG-PET and theother for MRI gray matter VBM) for univariate SPM respectively. Thespatial patterns are the singular image pair for the agnostic PLS andthe unmixed PET and MRI patterns separated out from the X-block for thenon-agnostic PLS. The p-values assessing the difference between the oldand young subjects were reported together for the agnostic PLS, SPM. Thepermutation results for non-agnostic PLS were reported and comparedamong three non-agnostic PLS analyses: X-block is PET only, X-block isMRI only, and X-block is both PET and MRI.

Results

Equivalence of Power Algorithm with MATLAB SVD:

The Power algorithm was implemented using MATLAB on an IBM A31 laptoprunning Linux operating system. The accuracy of the algorithm was testedagainst the MATLAB SVD implementation (svd.m and svds.m) using randomlygenerated matrices of varying sizes (100 by 100 up to 6500 by 6500). Itwas found that the implementation of power algorithm was equivalent toits MATLAB counterpart. However, for a computer with 1 GB RAM and 1 GBswap space, MATLAB svds.m crashed for a moderately large size matrix(6500 by 6500).

Numerical Similarities Between Agnostic Exact PLS and Inexact AgnosticPLS:

The inexact PLS implementation (i.e., with matrix size reduction first)was tested and compared to the exact solution again using randomlygenerated matrices of varying sizes. Though results by the two methodswere not identical non-surprisingly, there existed consistentcorrelations between them. Illustrating this consistency using component4 calculated from a randomly generated matrix pair with 29 as the numberof subjects and 200 as the number of voxels, FIG. 7 shows a very strongcorrelation between the exact method (x-axis, the column 4 of the matrixU) and the inexact method (y-axis, the column 4 of e^(T)U₁). Over 100simulated matrix pairs, however, the correlation coefficients fluctuatedbetween 0.73 and 0.99. It was also found that the first component waspoorly correlated when both matrices X and Y are non-negative (e.g.,uniformly distributed random univariate taking values over the interval[0 1]). In processing our real data, fortunately, both matrices weremean removed.

Individual SPM-PET, SPM-MRI Patterns and the Covarying PET/MRI Patternsof the Agnostic PLS:

The different analytical techniques were first compared by visuallyinspecting the changes, uncovered by each of them and shown in FIGS. 2A,6A, and 8A-8B, over the brain volume. FIGS. 2A, 6A, and 8A-8B are forexact and inexact agnostic PLS respectively. The left image on FIG. 8Ais for SPM-MRI (VBM) and the right image on FIG. 8A is for SPM-PET. Thevoxelwise bi-variate SPM is on FIG. 8B. Though they are of distinguishednature (intra-modality and voxel-wise univariate or voxel-wisebi-variate vs. inter-modality and multivariate) and should beinterpreted differently, overall they looked very similar to each other.

As an example, FIG. 2A displays agnostic PLS uncovered PET and MRIcovariance patterns superimposed on the standard brain anatomical map(left panel MRI gray mater singular image, right panel: PET singularimage). They were created with an threshold of p=0.05 to both positiveand negative values of the singular images after normalization by thebootstrap estimated standard errors (100 runs). The combined patternindicated consistent covarying lower gray matter concentrations andlower cerebral metabolic rate for glucose (CMRgl) occurred concurrentlyin medial frontal, anterior cingulate, bilateral superior frontal andprecuneus regions; posterior cingulate and bilateral inferior frontalregions were only seen (with negative pattern weights) on the PET butnot on the MRI; some white matter regions, caudate substructure, andoccipital regions were shown to be concurrently preserved or on the graymatter distribution alone. These cross-sectional young and old groupMRI/PET findings indicate that, in very healthy adults, age groupdifference is associated with regionally distributed and interlinkeddual network patterns of brain gray matter concentration and CMRglchanges. Unlike the individual SPM PET or SPM MRI/VBM analyses,inter-modality PLS quantified the linkage strength between the twodatasets (optimized latent variable pair covariance).

Linkage between FDG-PET and MRI gray matter concentration revealed byAgnostic PLS: The first overall linkage strength index is the squaredcorrelation coefficient for the latent variable pair, t and u, which isfound to be as strong as R2=0.73 (R=0.854 and p<3.77e-9) for theagnostic PLS. This close correlation is the basis for one to interpretthe MRI and PET covarying spatial patterns and their interactions. Foragnostic PLS, the overall explanatory power of X-block to eachindividual variable in Y-block was assessed by correlating the X-blocklatent variable t with y1, y2, y3, . . . in Y-block respectively. In ouragnostic PLS in the current study where MRI was treated as X-block, theoverall explanatory power is the overall anatomical influence on theFDG-PET spatial pattern (the map formed with correlation coefficients ofthe MRI gray tissue latent variable with the FDG-PET measurementvoxel-by-voxel). This is shown in FIG. 9A as corresponding t-score map(testing if the correlation coefficient is significantly different from0) with uncorrected p=0.001 threshold. In this map, the maximal positivecorrelation is in right middle cingulum at location (4 23 32), thecorrelation coefficient is 0.8 and p=1.97e-7 (T=6.917). The maximalnegative correlation is in the left putamen at location (−20 20 −8)R=−0.751 and p=2.63e-6 (T=−5.918). Similarly, FIG. 9B shows the PETlatent variable correlation with the voxel-wise MRI-gray tissueconcentration. The maximal positive and negative correlationcoefficients are 0.851 (p=4.92e-9, T=8.423) and −0.81 (p=9.96e-7,T=7.186) in right precentral at locations (50 −14 48) and right fusiform(34 −48 −4) separately. Note these two maps can be interpreted easily ascorrelation maps and are very similar to the singular images pair inFIG. 7.

In summary, FIGS. 9A-9B show spatial functional and structuralcorrelation maps. Shown in FIG. 9A is the correlation map between thePET measurements from each brain voxel and the global MRI gray matterlatent variable. Shown in FIG. 9B is the correlation map between the MRImeasurements from each brain voxel and the global PET latent variable.The significance is p=0.001 uncorrected for multiple comparisons.

Patterns uncovered in the non-agnostic PLS with both PET and MRI data inthe X-block: FIG. 10 shows the PET and MRI spatial distribution patternsthat contributed significantly in distinguishing young and old subjectsin non-agnostic PLS (i.e., the old young subject group membership is theMBPLS Y-block, and PET and MRI were both used in X-block as predictors).The overall similarities between the agnostic PLS singular images andthe non-agnostic results are apparent. That striking similarities speakfor the fact that the old young subject differences in our studies areoverwhelming and the dominant variations.

Distinguishing old and young subjects by various methods I: Agnostic PLSin comparison with SPM In contrast to SPM, inter-network agnostic PLScombines information from both modalities and provides a more powerful(with smallest type-I error) global index. For example, the multiplecomparison corrected type-I error for the global maxima is p=0.005(local maxima at [48 14 −2] in right insula, uncorrected p=2.37e-7) forPET and p=0.00001 ([50 −18 52] right postcentral, uncorrectedp=9.84e-12) for MRI separately. The multivariate (dual-variate in thisstudy) SPM multiple-comparison corrected type-I error is 1.34e-7(location [10 14 −14] right rectus, uncorrected p=1.06e-12). Note thatunivariate SPM-PET or SPM-VBM/MRI is one-tailed while that dual-variateSPM is two-tailed. In contrast, the significance is p=9.14e-11 andp=4.443e-12 for inexact and exact agnostic PLS respectively contrastingyoung and old subjects without the need to correct multiple comparisons(see FIG. 2A for exact agnostic PLS results).

In summary, FIG. 10 shows spatial functional and structural correlationmaps with cocavying MRI maps on the left and PET maps of the right,where the patterns were generated by non-agnostic PLS operations, andFIG. 2A is a graph showing the result that the total old and youngsubjects are separated into respective group by using the latentvariable pair obtained by an agnostic PLS operation.

As a further test for the power difference between exact and inexactagnostic PLS, the Jacknife analysis was used to examine the accuracy ofclassifying the subject who was left out at each of 29 runs. A linearclassifier was determined first in each run based on the information ofthe remaining 28 subjects. The classification was to assign the left-outsubject to the young or old group based on his/her PET and MRI latentvariable numerical values against the classifier. 100% accuracy wasobtained for the exact agnostic PLS procedure. For the inexact agnosticPLS, however, 3 of 29 subjects were misclassified (89.7% accuracy). Thiscomparison is very preliminary as only linear discrimination wasconsidered. Visual inspection of the data plot revealed the existence ofa non-linear discriminator which has yet to be further investigated.

Distinguishing old and young subjects by various methods II: Comparisonamong Non-agnostic PLS with PET only, MRI only, or PET and MRI together.Since the old/young group membership was actually the Y-Block innon-agnostic PLS, and the latent variable of X is formed to best predictthe membership, the resulted type-I error distinguishing the two groupshas to be estimated using non-parametric permutation. Out the total10000 random permutations with membership switched each time, twopermutations generated group membership distinction as strong as thenon-permuted one in non-agnostic PLS with PET alone (occurrence oftype-I error=2). For the non-agnostic PLS with MRI alone, the number oftype-I errors occurred 30 times. In contrast, the non-agnostic PLS withMRI and PET together serving as X-block, there is no additionaloccurrence of type-I error (except the non-permutated run). In otherword, p=0.0003, 0.0031 or 0.0001 for running non-agnostic PLS with PETalone, MRI alone, or MRI and PET together.

Computational Speeds: Currently, the exact agnostic PLS computation,using the iterative Power algorithm, took about 15-46 hours depending oninitial values used in the iteration and with some code optimization onan IBM A31 laptop (with 1 GB memory). See Discussion section for more onits practicality, our proposals and our on-going efforts that willsignificantly reduce the computation time.

For the inexact agnostic PLS and non-agnostic PLS, the computationalspeed is compatible to a routine SPM analysis.

Discussions

The use of PLS is proposed to investigate covarying patterns betweenmulti-imaging datasets. With this technique, for example, researcherscan seek the function and anatomy linkage information. In addition, itcan be used to combine information from multi-dataset agnostically fornon-agnostically for subsequent statistical inferences. PLS is one ofseveral tools which can be potentially used for studying theinter-modality multi-imaging datasets. Not only can it be used toexplore multi-datasets as a preliminary step for subsequent model basedand hypothesis oriented analysis (see (Rajah and McIntosh 2005) forexample as in intra-modality PLS), more importantly it can constructlatent variable based index which can be used to evaluate groupdifferences, longitudinal changes and potentially treatment effects.

Interpretation of the agnostic PLS covarying pattern of a given latentvariable pair requires both the good grasp of the PLS theories and thebio-physiological aspect of the research questions. As a demonstrated inthis study, the understanding of the agnostic PLS results can be helpedfrom the individual univariate SPM findings. In any case, however,fundamental differences between intra-modality univariate SPM andmultivariate inter-modality PLS should be noticed. Also, it is alsoworth noting that the covarying pattern was generated with the bootstrapestimated voxel-wise standard deviations as the normalization factors.The difficulty to interpret the dual covarying patterns is alsoassociated with the fact that the PLS seeks the maximal covariances,cov(t,u)=w′X′Yc with respect to w and c (subject to ∥w∥=∥c∥=1). Thus,the pattern c and w play similar role as the eigen-images of PCA-SSMwhich is with largest accumulated total variance for FDG-PET and MRIgray matter map respectively. Much more importantly, c and w establishedthe association between the two datasets (via the correlationcoefficient between the latent variable pair t=Σw_(i)x_(i) andu=Σc_(i)y_(i)) and enabled the subsequent explanatory power analysis forfurther understanding of the relationship between the two datasets.

The PLS approach commonly used in the neuroimaging field is actually thedual-block PLS which is the one adopted in our agnostic inter-modalityPLS. As seen in the method section, dual-block PLS is a special case,and more importantly, the foundation of the general multi-block PLS(MPLS) which can handle data from more than two datablocks. Ournon-agnostic PLS analysis in the current study in based on MPLS with theold/young group membership as the third (and dependent) dataset. With inmind the success of the multi-block PLS analysis in the field ofchemometrics (Lopes et al. 2002; Westerhuis et al. 1996; Westerhuis etal. 1997; Westerhuis and Smilde 2001), more general use of MPLS inneuroimaging study will be the topic of future studies (such as to dealwith triple imaging datasets with or without newly defined objectfunction).

One limitation of the Power algorithm for the agnostic PLS is itsrelatively high computational expense due to the rate of convergencethat also depends on the ratio between the first and the second largesteigenvalues (Press et al. 1992). Though efforts are undertaken in ourlab to dramatically reduce the computation time as described below,routine agnostic PLS can be accomplished overnights with its currentimplementation (note the speed is affected mainly by the number ofvoxels, and only slightly by the number of subjects/scans). This isacceptable in basic research settings where demand for immediate resultdelivery is not as important as the issues of power or sensitivity whichwe believe the agnostic PLS is of advantageous as demonstrated by thisstudy. Furthermore, the need to adjust settings and re-run the analysisis unlikely as all the pre-processing steps are standard and performedby SPM. In any event, we are continuously working on the efficientimplementation of the inter-modality agnostic PLS. In fact, our initialinvestigation on the use of the inexact agnostic PLS solution as initialvalues for the Power algorithm iterative procedure demonstrate that thespeed can be significantly improved for routine use. Moreover, otheralgorithms such as QR and Rayleigh quotient (Borga et al. 1997) withspecial consideration for implementation efficiency will be evaluated.Finally, we are in the process of evaluating the need to implement theagnostic PLS in a high performance computing system which is locallyavailable to us.

MRI aging findings: Though there is not direct evidence on thehippocampus formation (HF) differences between young and old subjects,it had been reported that head-size adjusted HF volume is stronglyassociated with delayed memory performance (Golomb et al. 1994). Murphyet al., reported (Murphy et al. 1992) that the caudate and lenticularnuclei were significantly smaller in older than younger men. Thissignificant difference remained when their volumes were expressed as aratio of cerebral brain matter volume (Murphy et al. 1992). Decline incaudate nuclei, (but not in lenticular nuclei), in anterior diencephalicgrey matter structures and association cortices and mesial temporal lobestructures, but no in lenticular nuclei, thalamus and the anteriorcingulate were also reported in a separate study (Jernigan et al. 1991).Temporal cortex was found unrelated to aging process but posteriorfrontal lobe volume (DeCarli et al. 1994). Changes that are certain:ventricle enlargement, HF, caudate, and lenticular nuclei

PET aging findings: Frontal metabolism measured with positron emissiontomography is shown to be decreased relatively to that in other corticalor sub-cortical areas, in a population of healthy elderly compared toyoung volunteers (Salmon et al. 1991). Using visual qualitativeassessment (Loessner et al. 1995), decreased cortical metabolism,particularly in the frontal lobes were found, but not in basal ganglia,hippocampal area, thalami, cerebellum, posterior cingulate gyrus andvisual cortex. With partial volume effects corrected, Melter et al.,found only true decline in regional cerebral blood flow in theorbito-frontal cortex (Meltzer et al. 2000).

To validate the newly introduced inter-modality PLS, the data from oldand young subject groups with largest possible differences were used inthis study. It is no surprising that all methods detected the groupdifferences with statistical significance levels that many neuroimagingresearchers desired to have in their studies. For hypothesis testing, itmight be pointless to distinguish p=1.34e-7 and p=4.44e-12. Inperforming power analysis in planning a new study, on the other hand,these differences may translate to reduced costs or increasedsensitivity. Moreover, with this first validation accomplished, it isour hope that PLS will be sensitive enough to pick up subtlegroup/condition differences which other methods might fail to detect.

In conclusion, the proposed inter-modality PLS method can be used toseek the direct linkage between two image datasets, and can be used toexamine group differences with increased power.

FIG. 11 is a flowchart of an exemplary implementation of a process 1100in which a link is made between datasets to accomplish a useful result.In step 1102, a plurality of datasets (D_(i), i=1−I) are acquired. Eachdataset is acquired either by an imaging modality or a non-imagingmodality that are performed upon each of a plurality of objects (O_(n),n=1−N). In step 1104, a linkage is found between D_(i) and D_(j), whereD_(i) and D_(j) are not the same modality. In step 1106, the linkagebetween D_(i) and D_(j) is reduced to an expression of a singlenumerical assessment. In step 1108, the single numerical assessment isused as an objective, quantified assessment of the differences andsimilarities between objects (O_(n), n=1−N).

The linkage found in step 1104 will preferably be found using a partialleast squares (PLS) operation, such as a dual block (DB) PLS operationor a multi-block (MB) PLS operation. The objects (O_(n)) can beanatomical human parts such as finger prints, organs or tissues (e.g.;brain, breast), body fluids, facial features, etc. Alternatively, theobjects (O_(n)) can be manmade, such as a manufactured electronicdevice. The measurements of these objects can be taken as indices, forexample indices related to various aspects of performance or indices formeasuring appearance.

Datasets (D_(i)) acquired in step 1102 can be an imaging modality or anon-imaging modality. Examples of an imaging modality include, but arenot limited to, ultrasound, different PET and single photon emissiontomography radiotracer methods, structural, functional,perfusion-weighted, or diffusion-weighted MRI, x-ray computedtomography, magnetic resonance spectroscopy measurements of N-acetylaspartic acid, myoinositol, and other chemical compounds,electroencephalography, quantitative electroencephalography,event-related potentials, and other electrophysiological procedures,magnetoencephalography, and combinations of the foregoing imagingmodality. Examples of a non-imaging modality include, but are notlimited to, an electrophysiological measurement, a biochemicalmeasurement, a molecular measurement, a transcriptomic measurement, aproteomic measurement, a cognitive measurement, a behavior measurement,and combinations of the foregoing.

A more particular example of process 1100 is seen in FIG. 12 as process1200. Process 1200 begins in step 1202 at which FDG-PET (PET_(n))datasets and MRI (MRI_(n)) datasets are acquired, where each dataset isacquired upon each of a plurality of human subjects (O_(n), n=1−N). Instep 1204, a linkage is found between FDG-PET and MRI for theirrespective datasets PET_(n) and MRI_(n), where the FDG-PET datasets(PET_(n)) and the MRI datasets (MRI) are pooled into composite datasetsfrom all of the individual human subjects. In step 1206, the linkagethat was found in step 1204 between FDG-PET and MRI is reduced to anexpression of a single numerical assessment. In step 1208, the singlenumerical assessment is used as an objective, quantified assessment ofthe differences and similarities between the human subjects (O_(n),n=1−N) or between subject groups (for example, the N subjects aredivided into two groups).

FIG. 13 is a flowchart of an exemplary implementation of a process 1300in which a link is made between dataset to accomplish a useful result.In step 1302, at each of a plurality of t_(j) (time=t_(j), j=1-2), aplurality of datasets (D_(i) ^(j), i=1−I, j=1, 2) are acquired. Eachdataset is acquired either by an imaging modality or a non-imagingmodality performed upon each of a plurality of objects (O_(n), n=1−N).In step 1304, a treatment of some kind is administered to each of theobjects O_(n) between time t₁ and time t₂. In step 1306, a linkage isfound between D_(i) ₁ ^(j) and D_(i) ₂ ^(j) at time t_(j), where D_(i) ₁^(j) and D_(i) ₂ ^(j) are not the same modality. In step 1308, thelinkage between D_(i) ₁ ^(j) and D_(i) ₂ ^(j) at t_(j) to is reduced toan expression of a single numerical assessment. In step 1310, the singlenumerical assessment is used as an objective, quantified assessment ofthe treatment effect, from time t₁ to time t₂, upon the objects (O_(n),n=1−N).

A more particular example of process 1300 is seen in FIG. 14 as process1400. Process 1400 begins in step 1402 at which, for each of a pluralityof t_(j) (time=t_(j), j=1-2), an acquisition is made of dual datasets(i) FDG-PET datasets (PET_(n) ^(j), j=1, 2), and (ii) MRI datasets(MRI_(n) ^(j), j=1, 2), where both datasets are acquired from aplurality of human subjects (O_(n), n=1-N). In step 1404, there is anadministration of a treatment (for example, a diet that is hypothesizedas slowing the human aging process) on each of the human subjects O_(n)between t₁ and t₂ (situation a), or on a subset of the human subjectsO_(n)(situation b) of the human subjects O_(n) between t₁ and t₂. Instep 1406, a linkage is found between FDG-PET and MRI for the dualrespective datasets (i) FDG-PET datasets (PET_(n) ^(j), j=1, 2), and(ii) MRI datasets (MRI_(n) ^(j), j=1, 2), where the FDG-PET datasets(PET_(n) ^(j)) and the MRI datasets (MRI_(n) ^(j)) are pooled intocomposite datasets from all of the individual human subjects. In step1408, indices of the linkage between FDG-PET and MRI at t_(j) arereduced to a single numerical assessment. In step 1410, a use is made ofthe single numerical assessment to assess the treatment effect, fromtime t₁ to time t₂, upon the human subjects (O_(n), n=1−N) (situationa), where the assessment represents a measurement of the differentialtreatment effects between a sub-set of the N human subjects and the restof the human subjects (situation a), or the assessment represents ameasurement of the effect of the treatment in comparison to a controlgroup that received no treatment (situation b).

While preferred embodiments of this invention have been shown anddescribed, modifications thereof can be made by one skilled in the artwithout departing from the spirit or teaching of this invention. Theembodiments described herein are exemplary only and are not limiting.Many variations and modifications of the method and any apparatus arepossible and are within the scope of the invention. One of ordinaryskill in the art will recognize that the process just described mayeasily have steps added, taken away, or modified without departing fromthe principles of the present invention. Accordingly, the scope ofprotection is not limited to the embodiments described herein, but isonly limited by the claims that follow, the scope of which shall includeall equivalents of the subject matter of the claims.

APPENDIX A: Iterative Power Algorithm Code, in Comparison to MATLABsvds.m

The MATLAB code for SVD calculation using power algorithm in comparisonto MATLAB routine svds.m is given below. Note both the example poweralgorithm code and svds.m need the whole matrix to be in memory. Inimplementing power algorithm in our PLS analysis, all the matrix byvector, vector by scalar multiplications are done by reading in onesub-matrix a time.

MATLAB code for SVD Calculation Using Power:

m=1000; n=1000; %x and y dim of the matrix (you can change) I=1; %onlyone singular value for demonstration purpose A=randn(m,n); %randomlygenerated matrix to test epsilon=1.0e−9; %convergence numIte=5000;%number of iterations %power algorithm below: u=randn(m,1);v=randn(n,1); %initials for iteration d=zeros(I);sigma=u′*A*v; fork=1:numIte; z=u; u=A*v; u=u/norm(u,2); v=A′*z; v=v/norm(v,2);sigma=u′*A*v; error=norm(A*v−sigma*u,2); ifsigma<0;u=−u;sigma=−sigma;end; %my own addition if rem(k,100)==0;disp(sprintf(‘%5d %9.7f %7.4f’,k,error,sigma)); end; iferror<epsilon;break;end; end sigma=u′*A*v;norm(A*v−sigma*u,2);norm(A′*u−sigma*v,2); %duplicate for fun :-) [q,sig,r]=svds(A,2);%conventional SVD by matlab %compare the power result with conventionalSVD if exist(‘h1’)~=1;h1=figure; h2=figure;set(h1,‘unit’,‘centimeters’);set(h1,‘pos’,[2 12 15 10]);set(h2,‘unit’,‘centimeters’);set(h2,‘pos’,[18 12 15 10]); set(h1,‘unit’,‘pixels’);set(h2,‘unit’,‘pixels’);end; figure(h1);%linrfit(u,q(:,1));title(‘u against q’); %my own use if(q(1,1)−q(100,1))/(u(1)−u(100))<0; plot(u,−q(:,1),‘o’);title(‘u againstq’); else;plot(u,q(:,1),‘o’);title(‘u against q’);end; figure(h2);%linrfit(v,r(:,1));title(‘v against r’); %my own use if(r(1,1)−r(100,1))/(v(1)−v(100))<0; plot(v,−r(:,1),‘o’);title(‘v againstr’); %45 degree line? else;plot(v,r(:,1),‘o’);title(‘v against r’);end;disp(sprintf(‘%7.4f %7.4f’,sigma,sig(1,1)));

B: Monte-Carlo Simulation Abstract

The abstract below was presented at the Nuclear Medicine annual meeting,June 2004. It reported our efforts in developing a Monte-Carlosimulation procedure to type-I error and statistical power calculationfor some newly proposed indices for neuroimaging studies taking multiplecomparison into consideration. This abstract described here is notspecially designed for PLS, but can be easily adopted for PLS.

A Monte-Carlo Simulation Package For The Calculation Of StatisticalPower, Familywise Type I Error Of Various Global Indices Associated WithNeuroimaging Studies, by Kewei Chen, Ph.D, Eric M. Reiman, MD, Gene E.Alexander, Ph.D, Richard D. Gerkin, MD, MS, Daniel Bandy, MS, thePositron Emission Tomography Center, Banner Good Samaritan MedicalCenter, Phoenix, Ariz.; the Department of Mathematics and Statistics,Arizona State University; the Departments of Radiology and Psychiatry,University of Arizona; the Department of Psychology, Arizona StateUniversity; and the Arizona Alzheimer's Research Center and theAlzheimer's Disease Core Center, Phoenix, Ariz., USA.

Introduction:

To account for the familywise type I errors in neuroimaging studies,various approaches have been successfully applied. Revisiting theMonte-Carlo concept, we developed such a simulation package introducingvarious new global indices indicative of brain functional changes.

Methods:

Package description: The simulation is performed over MNI space takingvarious experimental designs into consideration. Characterizing thestatistical parametric map as a whole, various new global indices wereintroduced that were related to conjunction of ‘lack of deactivation’and map-wise histogram shape or symmetry etc. These indices can serve asan activation index relevant to the research hypothesis and whose type Ierror theoretical calculations (either exact or approximate) are yet tobe realized. One example of the global indices is the ratio of thepositive maxima to the (absolute) negative maxima of the t-scores overthe brain volumes. Another is the kurtosis. In addition, the package cancalculate the type I error of study-specific (unusual) observations suchas the left/right symmetrical activation (not symmetry test), oractivation occurring only within a sub-brain region (at least one voxelwithin this sub-region is above a height threshold u, and no voxeloutside this region is higher than u1 (<<u)). This package is alsohelpful in examining the random field theorem (RFT) based p-value whenneeded (small sample size, low smoothness, etc.). Finally, this packagecan perform statistical power analysis taking the multiple comparisonsinto consideration. Example data: Oxygen-15 water PET data from 7subjects in a study of right hand movement was used to illustrate theuse of this computer package and the sensitivity of those globalindices.

Results:

With the settings identical to the SPM analysis of the example PET dataset, significant thresholds at p=0.05 as functions of the degree offreedoms (DF) were examined. It was found out, for example, thethresholds of the kurtosis of the map-wise histogram is a decreasefunction of DF, and behaves much like (3*DF-6)/(DF-4) plus a constant.To test the package ability for its power calculation, maximal effectsize of 5, 10 and 15% respectively for two-sample t-test with 32subjects in one group and 30 in another were introduced into thethalamus region with spatial variation. With multiple comparisoncorrected, the statistical powers were calculated to be 12, 68, and 98%respectively. For the example PET data set, it was found that thepackage performed equally well as or better than the RFT based approach.The hypothesized thalamus activation which did not survive the RFTcorrected p=0.05 was detected by several of the proposed indices, posthoc.

Conclusion:

The global features and the simulation package provide an alternative toevaluate exact type-I errors/statistical powers for neuroimagingstudies.

C: Preliminary Results of Alternative MBPLS

As stated in the Research Plan section, we assume there are m datasets,X₁, X₂, . . . , X_(m). tk is a latent variable representing X_(k) (k=1,2, . . . , m), t_(k)=Σw_(i) ^((k))x_(i) ^((k)) where x_(i) ^((k)) is theith column of matrix X_(k) and w_(i) ^((k)) is the corresponding weights(of unit norm). In this preliminary study, we tested the followingobject function for the calculation of the latent variablesmax(min_(k<l)(cov(t_(k), t_(l)))). MATLAB fmincon is used to optimizethis object function for obtaining the MBPLS solution with theconstrains that ∥t_(k)∥=1. In this testing, we used m=5 with number ofvariables being [10 15 8 20 15] for datablock 1 to 2 separately. Thenumber of measurements is 200. Multivariate Gaussian random numbers weregenerated for the five datasets as a whole with a zero vector as themean and an arbitrary positive-definite matrix (diagonal elements allequal one) as the covariance matrix. Once the datasets are generated,the MBPLS procedure with the newly defined objected function was run 50times each with different initial value (randomly generated).

We have the following conclusions from this preliminary numericalsimulation procedure (See FIG. 1):

I, Existence:

there exist t_(k)'s that are with very strong linkage (defined ascovariance) for all possible pairs as seen in FIG. 1. For example, thecorrelation coefficient between datablocks1 and 2 is 0.944 (the firstsubplot). In fact, the smallest absolute value among the pair-wisecorrelation coefficients is 0.929 (between datablocks 1 and 3).Furthermore, the existence was demonstrated by repeating the wholeprocedure many times with different number of datablocks and differentnumber of variables in each datablock.

II, Conditional Uniqueness:

As it is, the object function given above does not guarantee a uniquesolution. This is evidenced that the optimization process converged todifferent solutions when different initial values were given. In fact,the partial uniqueness exists in that m₁t_(k)'s are unique (regardlessof the initial values) for 1<m₁<m, and the rest m−m₁t_(k)'s are not. Tomake the solution unique, additional constrains are posted for the m−m₁datablocks. Let A be the index set for the datablock with unique t_(k)'sand Θ the one without. The optimization procedure is now to maximize:

min_(k<l,k,lεΛ∪Θ)(abs(cov(t _(k) ,t _(l))))+min_(kεΘ)var(t _(k)).

Providing different initial values (randomly chosen) at each of manyruns, we observed that the optimization procedure consistently convergedto a unique solution.

Note, when m=2, the solution of this procedure is equivalent to thefirst latent pair of the ordinary DBPLS.

III. PLS Implementation Via Other Algorithms

The core complexity in the conventional PLS implementation lies in thecomputation of eigenvectors for the latent variables u and v of Ω. Todate, numerous eigen reduction methods have been developed, includingGauss-Jacobi iteration, QR reduction, Arnoldi iteration, Lanczositeration, and Power algorithm, to name just a few. The Power iterationalgorithm that we implanted was in a sub-matrix approach as the matrixsize poses additional constrains. Every method has its own advantagesand issues. In our analysis, we plan to explore the power and QRalgorithms. In the power method a matrix whose eigenvalue needs to becomputed is multiplied by a starting vector, till convergence isobtained which is close to the eigenvalue. The rate of convergencedepends on the second larges eigenvalue. Since power iteration involvesrepeated matrix-vector products, which are easily implemented inparallel for dense or sparse matrix. While QR algorithm has been shownto be scalable for parallel computing machines, our objective would beto divide the matrix in to smaller units and compute at the nodeprocessor using the QR algorithm, review on the implementation is welldocumented [1,2,3]. In simplest form each iteration of QR methodrequires O(n³). It reduces to O(n²) if the matrix is in Hessenberg form,or O(n) if symmetric matrix is in tri-diagonal form. Preliminaryreduction is done by Householders or Givens transformation.

In addition to QR, we will also evaluate the use of the Rayleighquotient. Assuming the X-Y covariance matrix is C_(xy), the matrices, Aand B, to define the Rayleigh quotient are, respectively,

${A = {{\begin{pmatrix}0 & C_{xy} \\C_{yx} & 0\end{pmatrix}\mspace{14mu} B} = I}},$

where I is the identity matrix of the same size as matrix A. TheRayleigh quotient is defined as

${r(u)} = {\frac{u^{T}{Au}}{u^{T}{Bu}}.}$

It is known [6] that the global maxima point, u, of the function r(u)corresponds to

$u = \begin{pmatrix}{\mu_{x}w} \\{\mu_{y}c}\end{pmatrix}$

where w and c are the DBPLS solution, or the first singular image pair(and μx and μy are scalars so that ∥u∥=ρ, the covariance between t and uas defined in the DBPLS algorithm in the Background and Significancepart). Operationally, there is no need to form matrix A, Cxy or Cyx inadvance (which is extremely memory demanding). Instead and equivalently,we propose that the vectors u′X and Yu can be formal quickly at eachiteration step.

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What is claimed is:
 1. A computer-implemented method for assessing multi-modality datasets in the evaluation of a treatment for a disease, the method comprising: receiving at a first computing device a plurality of longitudinal datasets regarding each of a plurality of subjects, wherein a first longitudinal dataset D_(i) of the plurality of longitudinal datasets is a dataset of an imaging modality captured by an imaging system and received from the imaging system over a communications network, and a second longitudinal dataset D_(j) of the plurality of longitudinal datasets is a dataset of a non-imaging modality received from a second computing device over the network, the D_(i) and D_(j) datasets being independent datasets and forming respective square matrices; and executing instructions stored in memory of the first computing device, wherein execution of the instructions by a processor of the first computing device: partitions each of the D_(i) and D_(j) datasets into a plurality of submatrices, computes a linkage between the D_(i) and D_(j) datasets, wherein computing the linkage includes reading the D_(i) and D_(j) submatrices into memory one by one and processing the read submatrices one by one as needed to iteratively compute a single numerical assessment, wherein the single numerical assessment represents a differential effect of a treatment for a disease between at least two subsets of the plurality of subjects, and reports the single numerical assessment to a user of the first computing device at a display interface of the first computing device.
 2. The computer-implemented method of claim 1, wherein the imaging modality is selected from the group consisting of: ultrasound; positron emission tomography, single photon emission tomography radiotracer, or other nuclear medicine procedure; structural, functional, perfusion-weighted, or diffusion-weighted magnetic resonance imaging; x-ray computed tomography; magnetic resonance spectroscopy measurements of N-acetyl aspartic acid, myoinositol, or other chemical compounds; electroencephalography, quantitative electroencephalography, event-related potentials, or other electrophysiological procedures; magnetoencephalography; a medical imaging measurement procedure; and a non-medical imaging measurement procedure.
 3. The computer-implemented method of claim 1, wherein the non-imaging modality is selected from the group consisting of: a set of biochemical measurements; a set of molecular measurements; a set of genetic measurements; a set of transcriptomic measurements; a set of proteomic measurements; a set of cognitive measurements or clinical ratings; a set of behavioral measurements; and a set of measurements from a non-medical non-imaging modality.
 4. The computer-implemented method of claim 1, further comprising defining at least one of the D_(i) and D_(j) datasets based on a time interval.
 5. The computer-implemented method of claim 1, wherein the method further comprises administering a treatment for the disease to a set of the subjects over a period of time.
 6. The computer-implemented method of claim 1, further comprising generating, based on the single numerical assessment, a report on a severity level of a progression of the disease.
 7. The computer-implemented method of claim 1, wherein computing the linkage between the D_(i) and D_(j) datasets comprises using a partial least squares analysis.
 8. A computer-implemented method for assessing multi-modality datasets in the evaluation of a treatment for a disease, the method comprising: receiving at a computing device a plurality of longitudinal datasets regarding each of a plurality of subjects, wherein a first longitudinal dataset D_(i) of the plurality of longitudinal datasets is a dataset of a first imaging modality captured by one or more imaging systems and received from the one or more imaging systems over a communications network, and a second longitudinal dataset D_(j) of the plurality of longitudinal datasets is a dataset of a second imaging modality received from the one or more imaging systems over the network, the D_(i) and D_(j) datasets being independent datasets and forming respective square matrices; and executing instructions stored in memory of the computing device, wherein execution of the instructions by a processor of the computing device: partitions each of the D_(i) and D_(j) datasets into a plurality of submatrices, computes a linkage between the D_(i) and D_(j) datasets, wherein computing the linkage includes reading the D_(i) and D_(j) submatrices into memory one by one and processing the read submatrices one by one as needed to iteratively compute a single numerical assessment, wherein the single numerical assessment represents a differential effect of a treatment for a disease between at least two subsets of the plurality of subjects, and reports the single numerical assessment to a user of the computing device at a display interface of the computing device.
 9. The computer-implemented method of claim 8, wherein the first imaging modality is selected from the group consisting of: ultrasound; positron emission tomography, single photon emission tomography radiotracer, or other nuclear medicine procedure; structural, functional, perfusion-weighted, or diffusion-weighted magnetic resonance imaging; x-ray computed tomography; magnetic resonance spectroscopy measurements of N-acetyl aspartic acid, myoinositol, or other chemical compounds; electroencephalography, quantitative electroencephalography, event-related potentials, or other electrophysiological procedures; magnetoencephalography; a medical imaging measurement procedure; and a non-medical imaging measurement procedure.
 10. The computer-implemented method of claim 8, wherein the second imaging modality is selected from the group consisting of: ultrasound; positron emission tomography, single photon emission tomography radiotracer, or other nuclear medicine procedure; structural, functional, perfusion-weighted, or diffusion-weighted magnetic resonance imaging; x-ray computed tomography; magnetic resonance spectroscopy measurements of N-acetyl aspartic acid, myoinositol, or other chemical compounds; electroencephalography, quantitative electroencephalography, event-related potentials, or other electrophysiological procedures; magnetoencephalography; a medical imaging measurement procedure; and a non-medical imaging measurement procedure.
 11. The computer-implemented method of claim 8, wherein the method further comprises administering a treatment for the disease to a set of the subjects over a period of time.
 12. The computer-implemented method of claim 8, further comprising generating, based on the single numerical assessment, a report on a severity level of a progression of the disease.
 13. The computer-implemented method of claim 8, wherein computing the linkage between the D_(i) and D_(j) datasets comprises using a partial least squares analysis.
 14. A computer-implemented method for assessing a linkage between multi-modality datasets, the method comprising: receiving at a first computing device a plurality of longitudinal datasets regarding each of a plurality of subjects, wherein a first longitudinal dataset D_(i) of the plurality of longitudinal datasets is a dataset of an imaging modality captured by an imaging system and received from the imaging system over a communications network, and a second longitudinal dataset D_(j) of the plurality of longitudinal datasets is a dataset of a non-imaging modality received from a second computing device over the network, the D_(i) and D_(j) datasets being independent datasets and forming respective square matrices; and executing instructions stored in memory of the first computing device, wherein execution of the instructions by a processor of the first computing device: partitions each of the D_(i) and D_(j) datasets into a plurality of submatrices, computes a linkage between the D_(i) and D_(j) datasets, wherein computing the linkage includes reading the D_(i) and D_(j) submatrices into memory one by one and processing the read submatrices one by one as needed to iteratively compute a single numerical assessment, wherein the single numerical assessment represents a differential effect of a presence of a specified characteristic between at least two subsets of the plurality of subjects, and reports the single numerical assessment to a user of the first computing device at a display interface of the first computing device.
 15. The computer-implemented method of claim 14, wherein the characteristic is one or more of age, a weight measurement, a triglyceride measurement, or a body fat measurement.
 16. The computer-implemented method of claim 14, wherein the characteristic is the possession of a specified gene.
 17. The computer-implemented method of claim 16, wherein the gene is an apolipoprotein (APOE) gene.
 18. The computer-implemented method of claim 14, wherein the characteristic is the possession of at least one ε4 allele of the APOE gene.
 19. The computer-implemented method of claim 14, wherein the characteristic is the possession of two ε4 alleles of the APOE gene.
 20. The computer-implemented method of claim 14, wherein computing the linkage between the D_(i) and D_(j) datasets comprises using a partial least squares analysis. 